Ex: Estimate the Value of a Partial Derivative Using a Contour Map
Ex: Estimate the Value of a Partial Derivative Using a Contour Map

This sequence starts with an introduction to partial derivatives and continues through gradient. While some of the activities/problems are pure math, a number of other activities/problems are situated in the context of electrostatics. This sequence is intended to be used intermittently across multiple days or even weeks of a course or even multiple courses.

Shown below is a contour plot of a scalar field, $$\mu(x,y)$$. Assume that $$x$$
and $$y$$ are measured in meters and that $$\mu$$ is measured in kilograms.
Four points are indicated on the plot.

You are on a hike. The altitude nearby is described by the function $$f(x, y)= k x^{2}y$$, where $$k=20 \mathrm{\frac{m}{km^3}}$$ is a constant, $$x$$ and $$y$$ are east and north coordinates, respectively, with units of kilometers. You’re standing at the spot $$(3~\mathrm{km},2~\mathrm{km})$$ and there is a cottage located at $$(1~\mathrm{km}, 2~\mathrm{km})$$. You drop your water bottle and the water spills out.

Find the gradient of each of the following functions:

Consider the fields at a point $$\vec{r}$$ due to a point charge located at $$\vec{r}’$$.

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 around an infinite, uniformly charged, straight wire, starting from the following expression for the electrostatic potential: $$V(\vec r)=\frac{2\lambda}{4\pi\epsilon_0}\, \ln\left( \frac{ s_0}{s} \right)$$

You are watching: Gradient Sequence. Info created by Bút Chì Xanh selection and synthesis along with other related topics.