Gradient Examples

1. If \( \phi(x,y,z) = x^2y + zy + z^3 \), find \( \vec{\nabla}\phi(x,y,z) \).
2. Find the electric field when the potential is \( V = -\frac{1}{(x^2 + y^2 + z^2)^{1/2}} \).
3. A volcano has just erupted and lava is slowly flowing down from the mountain top. Suppose that the altitude of the mountain is given by \( z(x,y) = he^{-(x^2 + 2y^2)} \) where \( h \) is the maximum height, what is the direction that the lava flows at a given point \( (x,y) \)?

The volcano.