Static Fields: Fall-2025
HW 03 Practice: Due W2 D3

1. Vector Sketch (Rectangular Coordinates) Sketch each of the vector fields below.
   1. \(\boldsymbol{\vec F} =-y\,\boldsymbol{\hat x} + x\,\boldsymbol{\hat y}\)
   2. \(\boldsymbol{\vec G} = x\,\boldsymbol{\hat x} + y\,\boldsymbol{\hat y}\)
   3. \(\boldsymbol{\vec H} = y\,\boldsymbol{\hat x} + x\,\boldsymbol{\hat y}\)
2. Distance Formula in Curvilinear Coordinates

   The distance \(\left\vert\vec r -\vec r\,{}'\right\vert\) between the point \(\vec r\) and the point \(\vec r'\) is a coordinate-independent, physical and geometric quantity. But, in practice, you will need to know how to express this quantity in different coordinate systems.

   1. Find the distance \(\left\vert\vec r -\vec r\,{}'\right\vert\) between the point \(\vec r\) and the point \(\vec r'\) in rectangular coordinates.

      

   2. Show that this same distance written in cylindrical coordinates is: \begin{equation*} \left|\vec r -\vec r\,{}'\right| =\sqrt{s^2+s\,{}'^2-2ss\,{}'\cos(\phi-\phi\,{}') +(z-z\,{}')^2} \end{equation*}

      Hint: You may want to use the textbook: GMM: Change of Coordinates

   3. Show that this same distance written in spherical coordinates is: \begin{equation*} \left\vert\vec r -\vec r\,{}'\right\vert =\sqrt{r'^2+r\,{}^2-2rr\,{}' \left[\sin\theta\sin\theta\,{}'\cos(\phi-\phi\,{}') +\cos\theta\cos\theta\,{}'\right]} \end{equation*}

      Hint: You may want to use the textbook: GMM: Change of Coordinates

   4. Now assume that \(\vec r\,{}'\) and \(\vec r\) are in the \(x\)-\(y\) plane. Simplify the previous two formulas.

3. Power Series Practice
   1. Calculate the \(n=0, 1, 2, 3, 4\) coefficients of the power series for \(\cos{z}\) expanded around \(z=\pi\). Using these coefficients, find a power series approximation for this function.
      
   2. Plot both the original function and your approximation.
   3. For what values of \(z\) is your approximation “good”?
4. Series Notation 1

   Write out the first four nonzero terms in the series:

   1. \[\sum\limits_{n=0}^\infty \frac{1}{n!}\]

   2. \[\sum\limits_{n=1}^\infty \frac{(-1)^n}{n!}\]
   3. \begin{equation} \sum\limits_{n=0}^\infty {(-2)^{n}\,\theta^{2n}} \end{equation}

5. Series Notation 2

   Write (a good guess for) the following series using sigma \(\left(\sum\right)\) notation. (If you only know a few terms of a series, you don't know for sure how the series continues.)

   1. \[1 - 2\,\theta^2 + 4\,\theta^4 - 8\,\theta^6 +\,\dots\]

   2. \[\frac14 - \frac19 + \frac{1}{16} - \frac{1}{25}+\,\dots\]