Homework Help: Simple Gradient Question (funtion of two variables)

No, that is the gradient.

You are confusing this with the derivative "in the direction of unit vector v". That is equal to [itex]v\cdot \nabla f[/itex].

 

Yes, the gradient you calculated is correct.

In order to visualize the gradient in your problem,
depict a two dimensional graph and pick out arbitrary (x,y) points i.e. (2, 3) (0,1) (-1,0) (-1,-1) … and plug those into ∇(g). Use these corresponding vectors to depict the vector field. For example, at (1,0) the resulting vector would be <1,0>. Likewise…

(2,3) <2, -3>
(0,1) <0,-1>
(-1,0) <-1,0>
(-1,-1) ….
…. ….


To draw each vector, begin with the tail at the chosen (x,y) coordinates and move accordingly. For the first vector, begin at (2,3) then move in the positive x direction 2 units, then move down the negative y direction 3 units – there’s your endpoint. As you draw more vectors out, you should start to get an idea of the field’s behavior.