Maths for Statics

7/28/2019 Maths for Statics

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ALGEBRA

Areas of focus:

1. Expanding and factoring terms2. Solving an equation for an unknown3. Quadratic and biquadratic equations4. Logarithms and Exponents5. Determinants6. Systems of linear equations7. Solving linear systems in terms unknown variables8. Inequalities

Expanding and factoring terms:

The basic rules used for reorganization of terms in an expression are

The following relations are useful in either expanding or factoring simple algebraic

expressions.

x2 + Ax + B = (x + a)(x + b), where A = a + b, B = ab

Solving an equation for an unknown:
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The four basic operation of reorganization, regrouping, multiplication by a common

factor, and division by a common factor can be used to solve for a unknown variable

in an equation

Reorganization:

Reorganization is the process of moving the terms separated by addition, subtraction,

or equality signs. When reorganization moves a term from one side of equality to

another, then the sing of the term must be changed. The following is a simple example

of reorganization. All the equations convey the same information.

Regrouping:

The process of regrouping involves the gathering of term with common factors

together. This process frequently involves reorganization, expansion of term and

factoring of common expressions. The following is a simple example of regrouping of

terms.

Another example of regrouping is

Multiplication by a common factor:

One can multiply all terms in an equation by a common factor. The resulting equation

retains the relation between the variables if the factor is not zero or infinity. A simple

example of multiplication by a common factor is given by


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Division by a common factor:

One can divide all terms in an equation by a common factor. The resulting equation

retains the relation between the variables if the factor is not zero or infinity. A simple

example of division by a common factor is given by

Solving for a variable:

The process of solving for a variable combines the above operations to find the value

of a given variable in terms of the other variables in an equation. For example, the

value ofx in the following equation can be found by these steps:

Solving fory in terms ofx andzin this same equation can be done through the

following steps.

Quadratic and bi-quadratic equations:

The quadratic equation is given by the equation ax2 + bx + c = 0, wherex is the

variable to be solved for and a, b, and c are coefficients that do not depend onx. The

solution to the quadratic equation is known as its roots and can be evaluated by


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The roots of the quadratic equation are real numbers only when b2-4ac is positive or

zero.

The bi-quadratic equation is ax4 + bx2 + c = 0, and has the roots

Logarithms and Exponents:

Logarithms and exponents are related by the following relation:

Where a is the base of the logarithm. Ifa = e = 2.718282 then the logarithm is known

as the natural log, and ifa = 10 it is known as the common log. The natural log ofx is

also written as ln(x).

Some properties of the logarithm that hold for all bases are

log (xy) = log (x) + log (y)

log (x/y) = log (x) - log (y)

log (xn )= n log (x)

Some common values of the logarithm are

The argument of a logarithm can only be positive. The value of the logarithm is

negative for arguments between zero and one, and positive for arguments grater than

one (i.e., log(1) = 0, log (x) > 0 ifx > 1; log (x) < 0 if 0


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One can use these relations to help solve equations containing exponents and

logarithms. For example, one can solve forx by following the steps:

Determinants:

The determinant of a two by two matrix is given by

The determinant of a three by three matrix is given by

Systems of linear equations:

A system of linear equations is a set of two or more equations that are linear in the

designated variables. An example of a system of two linear equations in the

variablesx andy is

One can solve a system of linear equations in terms of the designated variables by one

of several different methods.

Elimination and back substitution:


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The most systematic method of solving a system of linear equations is by elimination

and back substitution. This involves elimination of one equation by taking one

equation and solving for one variable in terms of the other variables and then

substituting this expression into the remaining equations to reduce the number of

equations and unknown variables. Once the system is reduced to one equation and one

variable, back substitution is used to solve for the previously eliminated unknownvariables. For example, for the system of equations

one can solve forx from the first equation to get

Substitution into the second equation gives one equation iny that is

This can be solved fory to get

Substitution ofy back into (c) givesx as

Other methods of elimination:

One can frequently find shortcuts to solving systems of equations by adding or

subtracting a multiple of one equation to the another equation. For example, the first

equation in the system of equations given by

can be multiplied by 2 to get the new, but equivalent, system


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Subtraction of the second equation from the first eliminatesx to give

This can be solved fory to get

Any one of the above equations can now be used to getx. Substitution into the second

of the original equations gives

Cramer's rule:

Cramer's rule can be used to solve a linear system of equations. For example, consider

the linear system

The solution to this system is

As can be seen, the solution of each unknown is the ratio of two determinants. The

denominator consists of the determinant of the coefficients. The numerator consists of


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the determinant of a matrix that is constructed from the matrix of coefficients by

replacing one column by the column of constants from the right-hand-side of the

system of equations. For the first unknown, the first column of the matrix of

coefficients is replaced. For the second unknown, the second column of the matrix of

coefficients is replaced, and so on.

Cramer's rule can be used for any size system. If the determinant of the coefficients is

zero (the system is singular), then the system cannot be inverted and does not have a

unique solution. A system for which the right hand side is zero for all the equations is

called homogeneous and has a nonzero solution only when the matrix of coefficients

is singular.

Solving linear systems in terms of unknown variables:

Sometimes one needs to solve a system of equations in terms of a set of unknown

parameters. For example, consider the following system of equations.

This system is linear inx andy. One can solve forx andy in terms ofzby simply

manipulating the equations as ifzwas another constant. For example, elimination

ofx to solve fory in terms ofzwill involve the following steps:

Back substitution ofy into (*) givesx in terms ofz.

Inequalities:

Inequalities are manipulated in a similar manner to equalities with one exception. Thedirection of the inequality changes every time the equation is multiplied by a negative

number. For example, consider the inequality


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If both sides of the inequality are multiplied by -1, then we obviously get

As another example, consider solving forx in the inequality

The steps are

GEOMETRY

Areas of focus:

1. Area and perimeter of basic two-dimensional objects: triangles, rectangles, circles,trapezoids, parallelograms, and ellipses

2. Volume and surface area of three-dimensional objects: cubes, spheres, parallelepipeds,ellipsoids, and cones

3. Relation among angles when lines intersect4. Relation among angles when parallel lines intersect a line5. Relations between angle of a triangle6. Similar triangles7. Pythagorean theorem8. Pythagorean triplets9. Centroid of geometric objects10.Approximation of area and volume

Area and perimeter of basic two-dimensional objects:
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The following is a listing of the area and/or perimeter of a triangle, rectangle,

parallelogram, trapezoid, circle, sector of a circle, and ellipse.

Triangle:

b = base

h = height

Rectangle/Square:

Area = a b

Perimeter = 2 (a + b)

Parallelogram:

Area = a h

Perimeter =


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Area = constant for parallelograms with equal base

and equal height.

Trapezoid:

Circle:

r= radius

d = diameter =2r

Area =

Circumference =

Sectors of circles:


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Arc Length,

Angle measured in radians,

Sector area =

Ellipses:

Volume and surface area of three-dimensional objects:

The following is a listing of the surface area and/or volume of a cube, parallelepiped,

cylinder, sphere, cone, and pyramid.

Cubes and rectangular parallelepipeds:


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Volume = abc

Surface area = 2(ab+bc+ca)

Parallelepipeds:

A = Area of base

h = Height

Volume =Ah

Right Circular Cylinder:

Volume =

Lateral surface =

Spheres:

r= radius


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Cones:

l2 = r2 + h2

Pyramid or irregular cone:

Relation among angles when lines intersect:


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Opposing angles are equal when two straight lines intersect, and adjacent angles add

to 180o (i.e., ).

Relation among angles when parallel lines intersect a line:

When a line intersects parallel lines it makes identical angles with both lines.

Relations between angle of basic objects:

Interior angles of a triangle:

Exterior angles of a triangle:


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Interior angels of a parallelogram:

In a parallelogram opposite angles are equal

Also: 2A + 2B =360o

Similar Triangles:

The trianglesABCandADEare called similar triangles. The sides of two similar triangles areproportional and the angels are the same. The respective heights of these triangles are also

proportional to the sides.

Pythagorean Theorem:

The Pythagorean Theorem states that for a right triangle, as shown, there exists a relation

between the lengths of the sides given by


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a2 + b2 = c2

Pythagorean Triplets:

There are Pythagorean triples for the sides a, b, and c of a right triangle. The triplets

(a,b,c) are related through the relation a2 + b2 = c2, such as (3,4,5), (5,12,13) and

(7,24,25). All constant multiples of these triplets (e.g., (6,8,10) from (3,4,5)) also

create Pythagorean triplets. The following are examples of right triangles with sides

given in terms of Pythagorean triplets.

Centroid of geometric objects:

The centroid of an object is the geometric center of it. The centroid of the following

objects are indicated by a C.


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Approximation of area and volume:

Strips:

The area of a strip can be approximated as the length of the strip times its width if the

strip is of uniform width and if its width is small compared to its length.


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s = length

t= width

Slabs and shells:

The volume of a slab can be calculated as the surface area times the thickness if the

thickness is uniform. The volume of a shell can be approximated using this relation,

but the thickness must be small relative to the principal radii of curvature of the shell.

A = surface area

t= thickness

Volume =At

TRIGONOMETRY

Areas of focus:

1. Degrees versus radians2. Trigonometric functions3. Trigonometric relations between complementary angles4. Pythagorean Theorem5. The fundamental relation between sine and cosine6. The unit circle and visualizing the trigonometric functions
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7. Inverse of trigonometric functions8. Law of sines9. Law of cosines10.Values of trigonometric at specific angles11.Trigonometric identities12.

Curves of sine, cosine, and tangent13.Approximation for small angles: sine, cosine, and tangent

Degrees Versus Radians:

One revolution is 360o, and is also 2 radians. Thus, due to linear proportionality of

the two scales, the conversion fromx degrees toy radians is:

One radian is equal to 3.14159..., and is normally approximated by 3.14. The

following table gives equivalent angles in degrees, radians, and revolutions.

Degrees Radians Revolutions

0o

0 0

30o

45

o

60o

90o

180o

270o

360o

1

Trigonometric Functions:
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The trigonometric functions are namedsine, cosine, tangent, cotangent, secant, and cosecant. A

trigonometric function has one argument that is an angle and will be denoted " ". In writing the

trigonometric functions one uses the abbreviated forms: , , ,

, , and , respectively. Also, sometimes these are written as ,

, , , , and , respectively.

The value of each trigonometric function for an acute angle (


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In this figure, and are complementary angles, meaning . Examination of the

basic relation between the trigonometric functions and the sides of the triangle reveal the

following relations between the complementary angles and .

Since , we can also write:

Pythagorean Theorem:

The Pythagorean theorem states that for a right triangle, as shown, there exists a relation between

the length of the sides given be

a2 + b2 = c2

There are also Pythagorean triples for (a,b,c), such as (3,4,5), (5,12,13) and (7,24,25) sided

triangles, and all constant multiples of these triplets (e.g., (6,8,10)).

Fundamental Relations Among Trigonometric Functions:

From the Pythagorean Theorem of plane geometry we know that x2 + y2 = r2. This can be usedto derive a basic relation between the sine and cosine functions.


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TRIGONOMETRY FOR STATICSPART 2:

The Unit Circle and Visualizing Trigonometric Functions:

The fundamental relation suggests that the sine and cosine can be

visualized by using a circle of unit radius. To do this, draw a circle of unit radius, as shown in the

figure. Next draw a radial line from the center of the circle to the its arc and making a counter

clockwise angle with the horizontal axis as shown in the figure. The projection of this line

onto the horizontal axis is , the projection of this line onto the vertical axis is ,

and if the radial line is extended to intersect the vertical lineAB one can get as shown inthe figure.

From the unit circle one immediately discovers that the sine and cosine functions can havevalues from -1 to 1, and that the tangent can have any value form to .

One denotes the quadrants of the unit circle as shown in the figure. It can be seen that the sinehas positive value in the first and second quadrants, and negative value in the third and fourth


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quadrants. The cosine has positive value in the first and fourth quadrant and negative value in the

second and third quadrants. The tangent has positive value in the first and third quadrants and

negative value in the second and fourth quadrants.

The unit circle can also help one memorize the values of the trigonometric functions.

For example, at

At

At


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At

Inverse of Trigonometric Functions:

The inverse trigonometric functions are: arcsine, arccosine, and arctangent. For a specific

valuez, these are written as: , , . For example, the

function provides the angles that has . In a similar manner,

and , respectively, provide the angles for which and .

For example, means the angle for which the sine has a value of 0.5. Thus, one

solution is . Likewise, has a solution .

The inverse trigonometric functions are also written as sin-1, cos-1, and tan-1. For

example, is the same as . This contradicts the convention established for

positive exponents. Therefore, even though


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The inverse trigonometric functions are multi valued. For example, the

angles all satisfy the relation and are, therefore,

the solutions to . This can clearly be seen on the unit circle since the projection of

radial lines at 30o

and 150o

onto the vertical axis are the same.

On the unit circle the addition of 360o

to any angle results in a new radial line that falls on top of

the original radial line. Therefore, the value of any trigonometric functions at an angle is the

same as its value at . This is also true for the addition of any integer multiple of 360o

so

that, for example, for any integern.

Law of Sines:


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The law of sines states that

This can be shown by considering the triangles AXB and CXB in the following

figure. We have and , hence

or . In a similar manner one can show that .

Law of Cosines:

The law of cosines states that

This can be shown by considering the triangle BXC

that gives:


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a2 = p2 + (CX )2 = p2 + (b - AX)2

or a2 = p2 + b2 + (AX)2 - 2b(AX) (1)

Considering the triangle AXB one gets:

p2 + (AX)2 = c2 and

Substituting these into (1) one obtains:

The other relations are obtained in a similar manner.

GO TO PART 3

TRIGONOMETRY FOR STATICS

PART 3:

Values of Trigonometric Functions at Specific Angles:

For 0o

and 90o: These functions are limiting values that can be observed from the

drawing. As side y approaches 0 (zero), thex approaches r.
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For 30o and 60o:

For 45o:

Trigonometric Identities:

Basic identities:


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Half-angles:

Identities in terms of tan (/2):

where

Curves of Sine, Cosine, and Tangent:

Sine function:


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The sine function is an odd function since

Cosine function:

The cosine function is an even function since

Tangent function:

The tangent function is an odd

function since

Note that

as

Approximation of Trigonometric Functions at Small Angles:


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As a close approximation, when an angle (expressed in radians) is very small, we may use the

following approximations

replace with replace with replace with unity (i.e., )

ANALYTICAL GEOMETRY

Areas of focus:

1. Commonly used coordinate systems2. Equation of a straight line3. Equation of a circle4. Equation of conic sections

Commonly used coordinate systems:

Rectangular coordinates in 2-D:

The location of a point in a two-dimensional plane can be represented by a pair of numbers

representing the coordinates of the point in a rectangular coordinate system. For example, the

pointA in the figure has coordinates (x, y) in the rectangularx-y coordinate system shown. To

determine a coordinate one draws a perpendicular onto the coordinate axis. In this case, thearrows on the coordinate axis indicate that points to the right of the origin O on thex-axis are

positive and to the left are negative. In a similar manner, points above the origin on they-axis are

positive and below it are negative.
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Polar coordinates:

The location of a point in a two dimensional plane can be represented by a pair of numbers

representing the coordinates of the point in a polar coordinate system. For example, the

pointA in the figure has coordinates in the polar coordinate system shown. In this

coordinate systemrrepresents the radial distance from the reference point O to the

pointA and represents the angle the line OA makes with the reference lineOB. As indicated by

the arrow, the angle is positive if measured counter clockwise from OB and negative if measured

clockwise.


The location of a point in three-dimensional space can be represented by a triplet of numbers

representing the coordinates of the point in a rectangular coordinate system. For example,pointA in the figure has coordinates (x, y, z) in the rectangular coordinate system shown. To get

these coordinates one can drop a perpendicular line fromA onto thex-y plane to get pointB andthen draw perpendiculars onto thex- andy-axes.


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Cylindrical Coordinates:

The coordinates of a point in three-dimensional space can be represented in cylindrical

coordinates by the triplet ( ) as shown in the figure.

Spherical Coordinates:

The coordinates of a point in three-dimensional space can be represented in spherical

coordinates by the triplet as shown in the figure.


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Equation of a straight line:

The equation of a straight line in a plane is given in thex-y coordinate system by the set of

points (x, y) that satisfy the equation

where, as shown in the figure, a represents the slope of the line in terms of its rise divided by its

run, and b is they-coordinate of the point of intercept of the line and they-axis.

One can evaluate the equation of a line from any two points on it. For example, considerpointsA andB shown in the figure with coordinates (x1,y1) and (x2,y2), respectively. Since ACD

and ABE are similar triangles, we have

This can be put in the above format by selecting


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If x1 = 0, then, as can be seen from the figure,

Equation of a circle:

The circle centered at the origin of a rectangular coordinate system is given by the set of all

points (x,y) that satisfy the equation


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where, as can be seen in the figure, ris the radius of the circle.

In polar coordinates the equation of a circle is given by specifying the radial coordinate rto be

constant.

For a circle centered at point (x1,y1) and of radius r, the equation of the points on the circle is

given by

Equations of conic sections:

The ellipse, parabola, and hyperbola are conic sections. Their curves have the distinct

characteristic that each point on the curve is such that the ratio of its distance from a line knownas the directrix and a point known as the focus is a given constant. In the figure, Fis the

focus,AB is the directrix and, the constant e, known as the eccentricity, defines the conic section

and is given by


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The value of the eccentricity defines the conic shape.

e < 1 gives an ellipse

e = 1 gives a parabola

e > 1 gives a hyperbola

Equation of an ellipse:

The equation of an ellipse in the rectangularx-y coordinate system is given by

where a and b are half the lengths, respectively, of the major and minor axis.


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Equations of parabolas:

The basic equation of a parabola in a rectangularx-y coordinate system is given by

Depending on the sign ofa, this equation will result in one of the two following graphs.

A parabola that is rotate 90o

can be represented by the equation

The graph of this is shown in the following figure.


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The equation for parabolas moved from the origin to another point is given in the following

figures.

Equations of hyperbolas:

The equation of a hyperbola centered at the origin with its major axis on thex-axis is given by


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The figure shows the graph of this equation, where F1 and F2 are the focal points and the major

axis is the line that passes through these points.

The lengths of the major and minor axis of the hyperbola are given by 2a and 2b, respectively, asshown in the following figure.

If the center of the hyperbola is moved from the origin to a point with coordinates (x1,y1), thenthe equation of the hyperbola becomes

To get a hyperbola with major axis parallel to the vertical axis, one can changex andy in theequations.

A hyperbola with equal major and minor axis and with axis rotated 45o

from thex-y axes has an

equation

where the major and minor axes each have a length of . The figure shows this hyperbola.


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CALCULUS

Areas of Focus:

1. Differentiation2. Maximization and minimization3. Partial derivatives4. Integration5. Integration over a line6. Double integrals7. Integration over an area8. Centroid of an area9. Integrating differential equations

Differentiation:

The derivative of a function is a measure of how the function changes as a result of a change in

the value of its argument. Given the functionf(x), the derivative offwith respect tox is written

as or as , and is defined by

As shown in the figure, the derivative of the function f(x) at pointx gives the slope of the

function atx in terms of the ratio of the rise divided by the run for the line AB that is tangent to

the curve at pointx.
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The derivative of the functionf(x) is also sometimes written asf '(x). As shown in the figure,

one can also write the definition of the derivative as

The basic rules of differentiation are

The derivative of commonly used functions:

The following is a list of the derivatives of some of the more commonly used functions.


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The product rule for derivatives:

Consider a function such asf(x)=g(x)h(x) that is the product of two functions. The product rule

can be used to calculate the derivative offwith respect tox. The product rule states that

For example, to take the derivative off(x) = (2x+3) (4x+5)2

one can follow these steps


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The chain rule for derivatives:

Consider the functionf(U), where Uis a function ofx. One can calculate the derivative offwith

respect tox by using the chain rule given by

For example, to calculate the derivative off(U) = Un

with respect tox, where U = x2+a, one can

follow these steps

Maximization and minimization:

A point on a smooth function where the derivative is zero is a local maximum, a local minimum,

or an inflection point of the function. This can be clearly seen in the figure, where the functionhas three points at which the tangent to the curve is horizontal (the slope is zero). This functionhas a local maximum atA, a local minimum atB, and an inflection point at C.


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One can determine whether a point with zero derivative is a local maximum, a local minimum, or

an inflection point by evaluating the value of the second derivative at that point.

Given a smooth functionf(x), one can find the local maximums, local minimums, and inflections

points by solving the equation

to get all pointsx that have a derivative of zero. One can then check the second derivative foreach point to get the specific character of the function at each point.

Global maximum and minimum:


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The global maximum or minimum of a smooth function in a specific interval of its argumentoccurs either at the limits of the interval or at a point inside the interval where the function has a

derivative of zero. As can be seen in the figure, the function shown has a global maximum atpointA on the left boundary of the interval under consideration, a global minimum at B, a local

maximum at C, and a local minimum atD.

Partial derivative:

The derivative of a function of several variable with respect to only one of its variables is called

a partial derivative. Given a functionf(x,y), its partial derivative with respect to its first argument

is denoted by and defined by

Since all other variables are kept constant during the partial derivative, it represents the slope ofthe curve one obtains when varying only the designated argument of the function.

For example, the partial derivative of the function with respect tox is

evaluated by treatingy as a constant so that one gets


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The chain rule:

Consider a functionf[U(x), V(x)] of two arguments Uand V, each a function ofx. The chain rule

can be used to find the derivative offwith respect tox by the rule

For example, consider the functionf = UV2

where U =2x+3 and V=4x2. The derivative offwith

respect tox is given by

Integration:

The integral of a functionf(x) over an interval fromx1 tox2 yields the area under the curve of the

function over this same interval.

LetFdenote the integral off(x) over the interval fromx1 tox2. This is written as


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and is called a definite integral since the limits of integration are prescribed. The area under the

curve in the following figure can be approximated by adding together the vertical strips of

area . Therefore, the integral is approximated by

This approximation approaches the value of the integral as the width of the strips approaches

zero.

Indefinite Integrals:

A functionF(x) is the indefinite integral of the functionf(x) if

The indefinite integral is also know as the anti-derivative. Since the derivative of a constant iszero, the indefinite integral of a function can only be evaluated up to the addition of a constant.

Therefore, given a functionF(x) to be an anti-derivative off(x), the functionF(x) + C, where Cis

any constant, is also an anti-derivative off(x). This constant is known as the constant ofintegration and may be determined only if one has additional information about the integral.Normally, a known value of the integral at a specified point is used to calculate the constant of

integration.

The basic rules of integration are


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The indefinite integral of commonly used functions:

The following is a list of indefinite integrals of commonly used functions, up to a constant of

integration [ ]:


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Note: Remember to add a constant of integration. You evaluate the constant of integration byselecting the constant of integration such that the integral passes through a known point.

Relating definite and indefinite integrals:

To obtain the value of the definite integral knowing the value of the indefinite integral of thefunction, one can subtract the value of the definite integral evaluated at the lower limit of

integration from its value at the upper limit of integration. For example, if you have the indefinite

integral

Note that C, the constant of integration, cancels in the subtraction and need not be included. It is

common to sometimes use the notation

Change of variables:

Given a function U(x), one can use to change the variable of integrationfromx to U . The change of variables results in the rule


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For example, given the function we can write this function as

where U= ax+b and . Therefore, the integral offcan be evaluated by using thefollowing steps.

Change of variables for a definite integral is similar with an additional change in the limits of

integration. The resulting equation is

For example, given the function , we can write this function as

where U= ax and . Therefore, the integral offcan be evaluated by using the followingsteps.


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Integration by parts:

Given the functions U(x) and V(x), one can use integration by parts to integrate the following

integral using the relation

For example, to evaluate the integral

One can take and so that and . Using

integration by parts we get

Integration over a line:

The integralFof functionf(s) over lineAB, that is defined bys = 0 tos = l, is written as

.


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When either the domain of integration or the function is described in terms of another variable,

such asx in the figure, one can evaluateFby a change of variables to get

Depending on the format the information is provided in, it might be necessary to use the

Pythagorean Theorem to relate the differential line element along the arc of the curve tothex andy coordinates. For example, in the figure shown we can see that

The sign of the root must be selected based on the specifics of the problem under consideration.

For example, consider integrating the functionf(x,y) =xy2

over the straight line defined in thefigure from pointA to pointB. Direct integration, using the relations

would yield


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Double integral:

The double integral of functionf(x,y) first integrating overx and then integrating overy is given

by

The notation implies that the inner integral overx is done first, treatingy as a constant.

Once the inner integral is completed and the limits of integration forx are substituted into the

expression, the outer integral is evaluated and the limits fory are substituted into the resulting

expression. The rules of integration are the same as used for single integration for both theintegration overx and the integration over y. To integrate overy first and then overx, the

integration would be written as


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One can also write the integral limits without specifying the variable (i.e., without using "x=" and"y="). The orderdxdy ordydx clearly specifies what variable a specific limit is associated with.

Consider the following example of double integration of the functionf(x,y) =xy2.

Unlike the example, the limits of integration need not be constants. There will be no problem as

long as the inner integral is conducted fist and the limits are substituted into the resultingexpression before the outer integral is evaluated. For example, consider the following integral.


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Integration over an area:

The double integralFof the function over the areaA is written as


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This integral is the sum offdA over the areaA. EachfdA is the volume of the column with

base dA and heightf. Thus, the integral gives the volume under the surface . To

accomplish the summation that is represented by the integral, one needs to section the domain ofintegrationA into small parts (i.e., into many small dAs). As shown in the figure, the domain of

integration can be sections into rectangular sections, each with an

area . At the limit of small element sizes, the sum of these areaelements adds up to the original domain of integration.

The differential element of area dA in a rectangularx-y coordinate system is given byeitherdA=dxdy ordA=dydx. The difference between the two is the order of integration. If done

correctly, the value of the integration does not depend on this selection, yet the ease ofintegration may strongly depend on the choice fordA. IfdA=dxdy is selected, then for each valueofy an integration overx is conducted from the left limit of the domain to its right limit. This

process fills the domain with differential elements of area and is shown in the figure.


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As can be seen from the figure, the limits of integration ofx depend on the value ofy so that the

integral is written as

On the other hand, if one takes dA=dydx, the order of integration changes. In this case, for

eachx betweenx1 andx2, the argument is integrated overy from the lower limit of area to itsupper limit. This is shown in the figure.

In this case the integral can be written as


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For example, consider the integral off(x,y) =xy over the domain shown in the figure.

The integral can be defined as

Alternately, one can get the integral from


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As can be seen, the result of the integration does not change based on the selection of the order

of integration, yet the setup of the integrals does change.

Polar coordinates:

The differential element of area in polar coordinates is given by

This is a result of fact that the circumferential sides of the differential element of area have a

length of , as shown in the figure. Otherwise the integration process is similar torectangular coordinates.


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Centroid of an area:

The centroid of an area is the area weighted average location of the given area. For example,

consider a shape that is a composite ofn individual segments, each segment having anareaAi and coordinates of its centroid asxi andyi. The coordinates of the centroid of this

composite shape is given by

As can be seen, the location of each segment is weighted by the area of the segment and after

addition divided by the total area of the shape. As such, the centroid represents the area weighted

average location of the body. For a continuous shape the summations are replaced byintegrations to get


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whereA is the total area.

Centroid of common shapes:

The following figures show the centroid of some common objects, each indicated by a C.


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Integration of differential equations:

Differential equations are relations that are in terms of a function and its derivatives. There are

some methods for solving these equations to find an explicit form of the function. Some of the

simplest differential equations are of the form

The variables in such equations can be separated to get


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and then integrated to get

where Cis the constant of integration. A slightly more complicated differential equation is one of

the form

The variables in such equations can be separated to get

and then integrated to get

In general, if one can separate the variables, as was done in the two above examples, then one

can use the methods of integration to integrate the differential equation.

For example, consider the differential equation

The variables in this equation can be separated to give


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The result of integrating this expression is

where the constant of integration can be found knowing a point (xo,yo) that the

function must pass through. For this case

and the complete solution can be written as

If the variables cannot be separated directly, then other methods must be used to solve

the equation.

VECTOR METHODS

Areas of focus:

1. Vectors and vector addition
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2. Unit vectors3. Base vectors and vector components4. Rectangular coordinates in 2-D5. Rectangular coordinates in 3-D6. A vector connecting two points7.

Dot product8. Cross product

9. Triple product10.Triple vector product

Vectors and vector addition:

A scalar is a quantity like mass or temperature that only has a magnitude. On the otherhad, a vector is a mathematical object that has magnitude and direction. A line of

given length and pointing along a given direction, such as an arrow, is the typical

representation of a vector. Typical notation to designate a vector is a boldfaced

character, a character with and arrow on it, or a character with a line under it

(i.e., ). The magnitude of a vector is its length and is normally denoted

by orA.

Addition of two vectors is accomplished by laying the vectors head to tail in sequence

to create a triangle such as is shown in the figure.
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The following rules apply in vector algebra.

where P and Q are vectors and a is a scalar.

Unit vectors:

A unit vector is a vector of unit length. A unit vector is sometimes denoted by

replacing the arrow on a vector with a "^" or just adding a "^" on a boldfaced

character (i.e., ). Therefore,

Any vector can be made into a unit vector by dividing it by its length.


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Any vector can be fully represented by providing its magnitude and a unit vector

along its direction.

Base vectors and vector components:

Base vectors are a set of vectors selected as a base to represent all other vectors. The

idea is to construct each vector from the addition of vectors along the base directions.For example, the vector in the figure can be written as the sum of the three

vectors u1, u2, and u3, each along the direction of one of the base vectors e1, e2, and e3,

so that


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Each one of the vectors u1, u2, and u3 is parallel to one of the base vectors and can be

written as scalar multiple of that base. Let u1, u2, and u3 denote these scalar multipliers

such that one has

The original vectoru can now be written as


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The scalar multipliers u1, u2, and u3 are known as the components ofu in the base

described by the base vectors e1, e2, and e3. If the base vectors are unit vectors, then

the components represent the lengths, respectively, of the three vectors u1, u2, and u3.

If the base vectors are unit vectors and are mutually orthogonal, then the base isknown as an orthonormal, Euclidean, or Cartesian base.

A vector can be resolved along any two directions in a plane containing it. The figure

shows how the parallelogram rule is used to construct vectors a and b that add up to c.

In three dimensions, a vector can be resolved along any three non-coplanar lines. Thefigure shows how a vector can be resolved along the three directions by first finding a

vector in the plane of two of the directions and then resolving this new vector along

the two directions in the plane.


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When vectors are represented in terms of base vectors and components, addition of

two vectors results in the addition of the components of the vectors. Therefore, if the

two vectors A and B are represented by

then,

Rectangular components in 2-D:

The base vectors of a rectangularx-y coordinate system are given by the unit

vectors and along thex andy directions, respectively.

Using the base vectors, one can represent any vectorF as


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Due to the orthogonality of the bases, one has the following relations.


The base vectors of a rectangular coordinate system are given by a set of three

mutually orthogonal unit vectors denoted by , , and that are along thex,y,

andzcoordinate directions, respectively, as shown in the figure.


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The system shown is a right-handed system since the thumb of the right hand points in

the direction ofzif the fingers are such that they represent a rotation around thez-axis

fromx toy. This system can be changed into a left-handed system by reversing the

direction of any one of the coordinate lines and its associated base vector.

In a rectangular coordinate system the components of the vector are the projections of

the vector along thex,y, andzdirections.For example, in the figure the projections of

vectorA along thex, y, andzdirections are given byAx, Ay, andAz, respectively.


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As a result of the Pythagorean theorem, and the orthogonality of the base vectors, the

magnitude of a vector in a rectangular coordinate system can be calculated by

Direction cosines:

Direction cosines are defined as


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where the angles , , and are the angles shown in the figure. As

shown in the figure, the direction cosines represent the cosines of the

angles made between the vector and the three coordinate directions.

The direction cosines can be calculated from the components of the

vector and its magnitude through the relations


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The three direction cosines are not independent and must satisfy the

relation

This results form the fact that

A unit vector can be constructed along a vector using the direction

cosines as its components along thex,y, andzdirections. For example,

the unit-vector along the vectorA is obtained from

Therefore,


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A vector connecting two points:

The vector connecting pointA to pointB is given by

Aunit vector along the lineA-B can be obtained from


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A vectorF along the lineA-B and of magnitudeFcan thus be obtained from the

relation

Dot product:

The dot product is denoted by " " between two vectors. The dot product of

vectors A and B results in a scalar given by the relation

where is the angle between the two vectors. Order is not important in the dot

product as can be seen by the dot products definition. As a result one gets

The dot product has the following properties.


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Since the cosine of 90o is zero, the dot product of two orthogonal vectors will result in

zero.

Since the angle between a vector and itself is zero, and the cosine of zero is one, the

magnitude of a vector can be written in terms of the dot product using the rule

Rectangular coordinates:

When working with vectors represented in a rectangular coordinate

system by the components

then the dot product can be evaluated from the relation


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This can be verified by direct multiplication of the vectors and noting

that due to the orthogonality of the base vectors of a rectangular system

one has

Projection of a vector onto a line:

The orthogonal projection of a vector along a line is obtained by moving

one end of the vector onto the line and dropping a perpendicular onto the

line from the other end of the vector. The resulting segment on the line is

the vector's orthogonal projection or simply its projection.

The scalar projection of vectorA along the unit vector is the length of

the orthogonal projection A along a line parallel to , and can be

evaluated using the dot product. The relation for the projection is


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The vector projection ofA along the unit vector simply multiplies the

scalar projection by the unit vector to get a vector along . This gives

the relation

The cross product:

The cross product of vectors a and b is a vector perpendicular to both a and b and has

a magnitude equal to the area of the parallelogram generated from a andb. The

direction of the cross product is given by the right-hand rule . The cross product is

denoted by a " " between the vectors


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Order is important in the cross product. If the order of operations changes in a cross

product the direction of the resulting vector is reversed. That is,

The cross product has the following properties.


When working in rectangular coordinate systems, the cross product of

vectors a and b given by


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can be evaluated using the rule

One can also use direct multiplication of the base vectors using the

relations

The triple product:

The triple product of vectors a, b, and c is given by


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The value of the triple product is equal to the volume of the parallelepiped constructed

from the vectors. This can be seen from the figure since

The triple product has the following properties


Consider vectors described in a rectangular coordinate system as


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The triple product can be evaluated using the relation

Triple vector product:

The triple vector product has the properties

Maths for Statics

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