Winter 2009

MTH_254Vector Calculus I

Winter 2009

Kjersten Criss

Samuel House

Simplify:

The first thing we have to do is find some friendly simplified equations for the isotherms.

For

Simplify:Following very similar algebraic steps one can find the equations for the other two isotherms:

For For For

MAGICAlgebra!

1

Graphing the IsothermsIn this case the isotherms are all ellipses centered about the origin. One can easily find the length of the major and minor axes when looking at the equations:

The major axis of this ellipse has a length of 1/2 and the minor a length of 1/4.The major axis has a length of 1 and the minor has a length of 1/2.The major axis has a length of 2 and the minor has a length of 1.

1

1

2

2 2

3

3

3

Finding the Family of Orthogonal Trajectories

The orthogonal trajectories will have a slope that is the negative reciprocal of the slope of the isotherms. The first thing we need to do is find the slope of the isotherms.

The equations of the isotherms all have form:

Where C is some constant.

Because it will be much easier to use implicit differentiation and then solve explicitly for y, that is what we will do.

When one side of our equation is solved for a constant we can use the formula:

Finding the Family of Orthogonal Trajectories

The slope of the isotherms is

Therefore the slope of the orthogonal trajectories is

We need to integrate in order to find the a general equation for the trajectories:

Notice that this is a separable differential equation that we should all know how to solve.

Exponentiate both sides of the equation:

This is a general equation for the family of orthogonal trajectories.

Graphing the Orthogonal Trajectories

This problem only asks for a few orthogonal trajectories of your choosing. During the test we would recommend you pick easy ones.

Like when

Compute ∇T(x0,y0)Original Equation

22 4100),( yxeyxT

Take the partial derivatives:22 4200 yx

x xeT 22 4800 yx

y yeT

The -∇T(x0, y0) equation is defined by:

Compute ∇T(x0,y0)

)4

1,2

1(),( 00 yx

Plug in the initial position points:

22

22

)4

1(4)

2

1(

)4

1(4)

2

1(

)4

1(800

)2

1(200

)4

1,2

1(

e

eT

)16

1(4

4

1

)16

1(4

4

1

200

100)4

1,2

1(

e

eT

2

1

2

1

200

100)4

1,2

1(

e

eT

Why is this the fastest way?

Textbook:

We know that is maximum when ϴ=0. Clearly, is a minimum when ϴ=π

Cartoon: