Gradient of the scalar field

1. Aug 25, 2009

CSNabeel

1. The problem statement, all variables and given/known data
Calculate the gradient of the scalar field f(x,y) = x$$^{2}$$ - y$$^{2}$$ . Sketch the gradient for a few point on two straight lines y = x and y = -x on the plane and comment on the properties of the sketch.

2. Relevant equations

3. The attempt at a solution
So I worked out the gradient to be:

f = 2xi - 2yj

and then I did this for the point

x y x -y
-2 -2 -2 2
-1 -1 -1 1
0 0 0 0
1 1 1 -1
2 2 2 -2

but then I got confused on how to implement the gradient to this to do the sketch! Help would be much appreciated

2. Aug 26, 2009

re444

The sketch would be something like this:

In each point on the plane, (x,y), there is a gradient vector as you said, 2xi - 2yj . these vectors point to the direction in the function's domain, which the main function has the greatest increase in its value.

3. Aug 26, 2009

HallsofIvy

Staff Emeritus
At each (x,y) draw a vector having x-component 2x and y-component -2y. That is, go to the right 2x and down 2y (assuming x and y are positive, of course).