Activity 13.1.3.
(a)
In Figure 13.1.7 there are three sets of axes showing level curves for functions \(f\text{,}\) \(g\text{,}\) and \(h\text{,}\) respectively. Sketch at least six vectors in the gradient vector field for each function. In making your sketches, you don’t have to worry about getting vector magnitudes precise, but you should ensure that the relative magnitudes (and directions) are correct for each function independently.
Eight skew ellipses centered at the origin. The major axis of the ellipses is the line \(y=-x\text{.}\) Labels on the ellipses range from \(25\) to \(200\) in increments of \(25\text{.}\) The axes range approximately from \(-8\) to \(8\text{.}\)
Seven equally-spaced lines with slope \(1/2\) plotted on axes that range approximately from \(-8\) to \(8\text{.}\) The lines are labeled from \(-60\) to \(60\) increments of \(20\text{.}\)
Axes ranging from \(-3\) to \(3\text{.}\) In each quadrant, there are nested curves that appear like rounded squares far out and proceed toward circles in the middle. The curves are all labeled with either \(0\) or \(1\text{.}\)
(b)
Verify that \(\vF(x,y) = \langle 6xy,3x^2+9\sqrt{y}\rangle\) is a gradient vector field by finding a function \(f\) such that \(\nabla f(x,y) = \vF(x,y)\text{.}\) For reasons originating in physics, such a function \(f\) is called a potential function for the vector field \(\vF\text{.}\)
(c)
Is the function \(f\) found in part b unique? That is, can you find another function \(g\) such that \(\nabla g(x,y)= \vF(x,y)\) but \(f\neq g\text{?}\)
(d)
Is the vector field \(\vF(x,y) = 6xy\vi +(2x+9\sqrt{y})\vj\) a gradient vector field? Why or why not?
