1. ## find the gradient and plot on level surface

$\displaystyle f(x,y,z) = z-x+y$ find $\displaystyle \nabla f(0,0,1)$ plot it on the level surface of f that passes through $\displaystyle (0,0,1)$

$\displaystyle \nabla f = -\hat{i} + \hat{j} + \hat{k} = <-1,1,1>$

$\displaystyle \nabla f (0,0,1) = <-1,1,1>$

and $\displaystyle f(x,y,z)$ is a plane passing through $\displaystyle (0,0,1)$

$\displaystyle z-x+y=1$ find intercepts and draw then plot from (0,0,1) the vector <-1,1,1>

??

2. ## Re: find the gradient and plot on level surface

A "level surface" for f(x,y,z) is a surface on which f(x,y,z)= C for some constant C. For f(x)= z- x+ y, you are correct, that will be a plane: z- x+ y= C. And, of course, (0, 0, 1) satisfies 1- 0+ 0= 1 so the level surface containing (0, 0, 1) is the plane z- x+ y= 1. Yes, grad f, at (0, 0, 1) is <-1, 1, 1> which does NOT lie in the plane z- x+ y= 1. So I don't know what this question is asking! Perhaps it means the projection of the vector on that plane.

3. ## Re: find the gradient and plot on level surface

yeah I may have to clarify that with my professor. When I sketched it, it made no sense.

4. ## Re: find the gradient and plot on level surface

gradient is orthogonal to the level surface.