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Question

Expect us to ask questions before we rate your response and thanks for the help!

This first section is an introduction to the project, not actually the question.

A vegetation disturbance index J(x,y) measures the vegetation loss for each point (x,y) in the landscape. If J(x,y) is large then there had been much disturbance and if it is small, there has been little disturbance. The function J(x,y) is a smooth function of x and y. (A smooth surface is a function that has a tangent plane at each point and for which the direction of the normal is a continuous function of the point of tangency.) You may assume that J has first and second order continuous partial derivatives.

Here are the instructions:

Your goals in this project consist of the following four steps:

1. Show that if the vegetation disturbance index J(x,y) is a smooth function of x and y, then Jx (Partial derivative) does not equal M and Jy (Partial Derivative) does not equal N for the M and N given below:

M(x,y) = 3.4*e^x*(y-7.8)^2 and N(x,y) = 22*sin(75-2*x*y)

It is impossible for the first partials of a smooth function J(x,y) to be equal to the M and N defined in the equation (1).

2. Find a condition that the correct M and N must satisfy in order for Jx = M and Jy = N, where the total differential of J is

dJ(x,y) = Jx(x,y)dx+Jy(x,y)dy = M(x,y)dx+N(x,y)dy

3. After you have defined the condition that M and N must satisfy, describe a method to find a formula for J(x,y) (solve for J given Jx = M and Jy = N) from the correct functions M and N.

4. Finally, provide an example of specific functions M(x,y) and N(x,y), nontrivial and interesting functions of x and y, to show that

(i) M and N satisfy the condition you derived

(ii) Find a formula for J(x,y) (up to an arbitrary constant).

Explanation / Answer

1.Note that since J is smooth Jxy = Jyx

   (J smooth means both Jxy and Jyx are continuous and thus by Clairaut theorem they are equal) So we should have

    My =     Nx (partial derivative of M wrt y and partial derivative of N wrt x s

    Given M and N cant be candidates because My and Nx are not equal as can be verified directly.

2. The condition to be satisfied is My = Nx (as talked about in part 1)

3. If M and N satisfy the above condition

   Jx = M , Jy = N

   J = integral (M) wrt x + f(y) (note we are integrating wrt x so constant of integration is a function of y)

   Also J = integral (N) wrt x + g(x) (note we are integrating wrt x so constant of integration is a function of y)

Equate the above two to obtain J = L(x,y) + constant

4. example M = y , N = x

   (i) My = 1 = Nx.Thus these M and N satisfy condition mentioned in 2

   (ii) applying method described in 3

       J = xy+ f(y) = xy + g(x)

       So f(y) and g(x) must both be equal to the same constant

      Hence J = xy+C

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