### multivariable calculus - (NOT a physics question) Is electric field always asymptopic to $x^{\alpha}$ for some rational $\alpha$?

In three dimensional space with origin $O$, you pick a finite number of points $P_1, P_2, \cdots, P_n$. To each point $P_i$ you assign a nonzero integer (positive or negative) $q_i$. For all other points $R$ in the plane, define the vector valued function $$\displaystyle \vec{F(R)} = \sum_{i = 1}^{n} \frac{q_i}{D(P_i, R)^2} \vec{r_i},$$ where $D(P_i, R)$ is the Euclidean distance between $P_i, R$, and $r_i$ is a vector of unit magnitude directed from $P_i$ to $R$. Now you pick a ray $\vec{\ell}$ originating from $O$ in any direction. Is it tru...Read more

### reference request - A better resource for vector calculus than Stewart?

I took the Math GRE Subject Test last October and tried to relearn vector calculus via the current edition of Stewart's text.I thought, to put it lightly, that the exposition was atrocious and unmotivated, with too much of a focus on memorizing the equations needed to solve problems. I did do well on the calculus questions on the Subject Test [I think], but if I need to teach myself vector calculus again, I can't use Stewart to do it.What do you all recommend for a good rigorous, motivational text on vector calculus? I need the book to cover Gr...Read more

### reference request - Multivariable calculus book with interesting examples and exercises

I'm studying Stewart's multivariable calculus book and I find most the examples and exercises pretty easy and repetitive. I would like to know if there are books in multivariable calculus with more intriguing and interesting examples. Just to make a comparison with one-variable calculus, I find Spivak's calculus book more interesting with less repetitive exercises than standard books.Remark: Just to be clear, I'm requesting for a book which doesn't necessarily need to be more theoretical, just having more interesting examples....Read more

### Multivariable calculus: hard problems with solutions

I'm practicing for my multivariable calculus exam and I'm having some trouble mostly because I have no way of knowing if my solutions are correct or not.For example, a typical problem goes like this:Let $f:\mathbb{R^2}\longrightarrow\mathbb{R}$ defined by:$$f(x,y)=\begin{cases}\sin(y-x) & \text{for} & y>|x| \\ \\0 & \text{for} & y=|x| \\ \\\frac{x-y}{\sqrt{x^2 + y^2}} & \text{for} & y<|x| \end{cases}$$Study $f$ with respect to continuity on its domain.Study $f$ with respect to differentiability on its domain.I th...Read more

### multivariable calculus - Need some help with computing line integrals for vector fields

I am a little confused on the computation of a Line Integral of a Vector Field.Here is what I have so far: $$\int_C \mathbf F \cdot d \vec r$$ (F is a vector field of n dimensions ($$n \ge 2- dimensions$$)I know that you need a parametrization of the Curve C(c=[{x(t),y(t)} $\in$ a $\le$ t $\le$ b])and that ||d$\vec r$||= $\sqrt{(dx/dt)^2+(dy/dt)^2} dt$But other then that I am a tad bit lost :P, So any help would be appreciated....Read more