Plotting Gradients of Scalar Fields on the Plane

[jlmac2001]

Here's the question:

Suppose we have a scalar field phi(r)=xy defined on the plane (x,y). Find th vector field gradient phi, and plot the result in a single graph atthe four points (1,1), (-1,1) (1,-1), (-1,-1) and at the four points (1,0),(0,1),(-1,0) and (0,-1).

Ans.

For the gradient of phi(r)=xy, I got yi + xj. Is this right?

 

[jlmac2001]

one more questio

how will I plot it. Do i have to solve the eequation at each of those points and plot the answer that I get?

 

[quantumdude]

Originally posted by jlmac2001
For the gradient of phi(r)=xy, I got yi + xj. Is this right? [/B]

Yes.

 

[quantumdude]

Originally posted by jlmac2001
how will I plot it. Do i have to solve the eequation at each of those points and plot the answer that I get?

No, it's not that involved. For each of the 4 points, you plug in the values of x and y (edit: plug them into the gradient). From those, you will get 4 vectors, which you need to sketch on the graph.

 

[jlmac2001]

is this right ?

Okay. So at (1,1) it would be i + j
at(-1, 1)= -i
at (1,-1) = i-j
at (-1,-1) = -i-j

How would I plot it?

 

[quantumdude]

Originally posted by jlmac2001
Okay. So at (1,1) it would be i + j

Right.

at(-1, 1)= -i
at (1,-1) = i-j

These are wrong. Remember that for your vector v, grad(v)=yi+xj.

at (-1,-1) = -i-j

This one is OK.

How would I plot it?

Go to the point in question and let that be the origin of a vector whose components are what you calculated them to be. For instance, in the first case go to the point (1,1) and draw the vector i+j.

Also, don't forget to do the same for the other 4 points listed. You will have 8 vectors in all.