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## Homework for Symmetries

- (DirectionalDerivative)
Imagine you're standing on a landscape with a local topology described by the function $f(x, y)= k x^{2}y$, where $k=20 \frac{m}{km^3}$ is a constant, $x$ and $y$ are east and north coordinates, respectively, with units of kilometers. You're standing at the spot (3 km,2 km) and there is a cottage located at (1 km, 2 km). At the spot you're standing, what is the slope of the ground in the direction of the cottage? Plot the function $f(x, y)$ in Mathematica. Does your result makes sense from the graph?

- (GradientPractice)
Find the gradient of each of the following functions:

$$f(x,y,z)=e^{(x+y)}+x^2 y^3 \ln \frac{x}{z}$$

$$\sigma(\theta,\phi)=\cos\theta \sin^2\phi$$

$$\rho(s,\phi,z)=(s+3z)^2\cos\phi$$

- (GradientPtChargeOrigin)
The electrostatic potential due to a point charge at the origin is given by: $$V=\frac{1}{4\pi\epsilon_0} \frac{q}{r}$$

Find the electric field due to a point charge at the origin as a gradient in rectangular coordinates.

Find the electric field due to a point charge at the origin as a gradient in spherical coordinates.

Find the electric field due to a point charge at the origin as a gradient in cylindrical coordinates.