The Games children play: the foundation for mathematical learning
Lec 2 Mathematical Foundation
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Transcript of Lec 2 Mathematical Foundation
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STRUCTURAL ENGINEERING
Mathematical Foundation
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Tensor, Matrices, Vectors
Tensor
Is a general name for physical measure (quantity) thatis generally described by a whole set of values and
these values may be dependent on space coordinates. Tensors are specified by their rank or order depending
on the no. of components they posses
In 3D- space a tensor of a rank N has 3N components
0th
order : 1 Component (Scalar) 1st order: 3 Components (Vector)
2nd order: 9 Components (Matrix)
3rd order: 27 Components (Tensor)
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Tensors are geometric entities introduced into
mathematics and physics to extend the notion of
scalars, (geometric) vectors, and matrices
Because they express a relationship between
vectors, tensors themselves are independent of a
particular choice of coordinate system
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Types of Notations
1- Matrix Notation
2- Index Notation
1- Matrix Notations
A matrix is a rectangular arrangement of numbers.
For example,
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The horizontal and vertical lines in a matrix are
called rows and columns, respectively
The numbers in the matrix are called its entries or its
elements
To specify a matrix's size, a matrix with m rows and
n columns is called an m-by-n matrix or m n
matrix, while m and n are called its dimensions In Example the Matrix is a 4x3 matrix
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More than one index is used to describe arrays or
number with two or more dimensions, such as the
elements of a matrix. The (i, j)th entry of a matrix A
is most commonly written as ai,j
where the first subscript is the row number and the
second is the column number
Free index & Dummy Index
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Free index
Must Appear ONCE in each term of expression or
equation
Dummy Index
Appears TWICE on a term can be replaced by
alternating index
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Matrix Addition
[A] + [B] = [C]
Ai,j + Bi,j = Ci,j
The sumA
+B
of two m-by-n matricesA
andB
iscalculated entry wise: (A + B)i,j = Ai,j + Bi,j
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Scalar Multiplication
The scalar multiplication cA of a matrix A and a
number c is given by multiplying every entry of A
by c
(cA)i,j = c Ai,j
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Transpose of Matrix
The transpose of an m-by-n matrix A is the n-by-m
matrix ATformed by turning rows into columns and
vice versa
(AT)i,j = Aj,I
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Matrix Multiplication
Multiplication of two matrices is defined only if the
number of columns of the left matrix is the same as
the number of rows of the right matrix
If A is an mxn matrix and B is an nxp matrix, then
their matrix productAB is the mxp matrix whose
entries are given by dot-product of the
corresponding row ofA
and the correspondingcolumn of B
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Matrix multiplication satisfies the rules (AB)C =A(BC) (associativity) and
(A+B)C = AC+BC as well as C(A+B) = CA+CB
matrix multiplication is not commutativeAB BA
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Identity Matrix
The identity matrix In of size n is the nxn matrix in
which all the elements on the main diagonal are
equal to 1 and all other elements are equal to 0
It is called identity matrix because multiplication
with it leaves a matrix unchanged
MIn = ImM = M
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Symmetric Matrix
In linear algebra, a symmetric matrix is a square
matrix that is equal to its transpose
The entries of a symmetric matrix are symmetric
with respect to the main diagonal
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Row & Column Matrix
In linear algebra, a row vector or row matrix is a
1 n matrix, that is, a matrix consisting of a single
row
The transpose of a row vector is a column matrix or
vector
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Scalar Matrix
A diagonal matrix with all its main diagonal entries
equal and rest are zeros is a scalar matrix
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Trace of a Matrix
In linear algebra, the trace of an nxn square matrix
A is defined to be the sum of the elements on the
main diagonal
Let Tbe a linear operator represented by the matrix
Then tr (T) = 2 + 1 1 = 2.
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Determinant of Matrix
In algebra, the determinant is a special number
associated with any square matrix
The determinant of a matrix A, is denoted det(A),
or without parentheses: det A
denotes the determinant of the matrix
has determinant det A = ad bc
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Inverse of Matrix
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Orthogonal Matrix
In linear algebra, an orthogonal matrix is a square
matrix with real entries whose columns (or rows) are
orthogonal unit vectors If [A]-1 = [A]T then [A] is orthogonal
If det [A] = +1 then [A] is Proper orthogonal
If det [A] = -1 then [A] is Improper orthogonal
For orthogonal Matrix [A][A]T = [I]
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Vector
Physical quantities which have both magnitude anddirection, such as force, in contrast to scalarquantities, which have no direction.
Where vx, vy, and vz are the magnitudes of thecomponents of v.
Where v1, v2, , vn 1, vn are the components ofv.
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Dot Product
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Cross Product
The cross product of two vectors a and b is denoted
by a b
a = a1i + a2j + a3k = (a1, a2, a3)And
= b1i + b2j + b3k = (b1, b2, b3).
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Simultaneous Equations
In mathematics, simultaneous equations are a set
of equations containing multiple variables
This set is often referred to as a system of
equations
A solution to a system of equations is a particular
specification of the values of all variables that
simultaneously satisfies all of the equations
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Solution of Equation
Solving this involves subtractingx + y= 6 from
2x + y= 8 (using the elimination method) to remove
the y-variable, then simplifying the resulting
equation to find the value ofx, then substituting the
x-value into either equation to find y.
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Solution of Simultaneous Equations
Methods of Solution of Simultaneous Equations
1- Cramers Rule
2- Gauss Elimination Method
3- Gamss-Seidel Iteration
4- Choleskys Method
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Flexibility Method
In structural engineering, the flexibility method is
the classical consistent deformation method for
computing member forces and displacements in
structural systems
Its modern version formulated in terms of the
members' flexibility matrices also has the name the
matrix force method due to its use of member
forces as the primary unknowns
Flexibility is the inverse of stiffness
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The relationship between actions and displacements play an important role
structural
analysis. A convenient way to see this relationship is through a linear, elastic
spring
The action A will compre
ss (translate
) the
spring an amount D.T
his can be
expressed
through the simple expression:
In this equation F is the flexibility of the spring, and this quantity is defined as the
displacement produced by a unit value of the action A.
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This relationship can also be expressed as
A = KD
Here K is the stiffness of the spring and is defined as the action required toproduce a unit
displacement in the spring. The flexibility and stiffness of the spring are
inverse to one another.