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

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

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

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

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

Let $V$ be a convex region in $\mathbb R^3$ whose boundary is a closed surface $S$ and let $\vec n $ be the unit outer normal to $S$. Let $F$ and $G$ be two continuously differentiable vector fields such that $\mathop{\rm curl} F=\mathop{\rm curl} G; \mathop{\rm div} F=\mathop{\rm div}G$ everywhere in $V$ and $G\cdot\vec n = F\cdot\vec n$ every where on $S$. Then is it true that $F=G$ everywhere in $V$? I tried as follows: Let $H=F-G$ , then $\mathop{\rm curl} H=O$, so there is a scalar field $f$ such that $H=\nabla f$, and also $\mathop{\rm ...Read more

I have this identity which I am trying to show; div($\textbf{u x v}$)= $\textbf{v} \cdot \text{curl}(\textbf{u}) - \textbf{u} \cdot \text{curl}(\textbf{v})$;but have also come across, $\textbf{a} \cdot (\textbf{b x c}) = \textbf{c} \cdot (\textbf{a x b})$; and so using this identity why is it not true that,div($\textbf{u x v}$)= $\textbf{v} \cdot ( \nabla \textbf{ x u})=\textbf{v} \cdot$ curl($\textbf{u})$?My question is why doesn't the last identity I have written hold? I have been told it is something to do with the fact that $\nabla$ is an o...Read more

I am currently studying Calculus on Manifolds .I am studying Spivacks book along with Munkres and J.Shurmans notes since i might not understand something from one book to another.What i noticed is so far no one mentions a way to calculate the total derivative of a function f except using the definition ."finding a linear(multilinear) map such that a limit exists." What i wanted to ask is that if we know The Matrix of this map (jacobian matrix) and the Basis of the Vector Spaces .Then i can find that map=the total derivative with the standard ...Read more

The question is: Prove that the volume bounded by the ellipsoid $\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1$ is $4\pi abc/3$ using the change of variables formula. Let $x=a\rho $sin$\varphi $cos$\theta$, $y=b\rho $cos$\varphi $cos$\theta$, $z=c\rho $cos$\varphi $, then we can use the change of variables formula. However, the answer says the Jacobian of the change of variables $x$, $y$, $z$ is $abc\rho^2 $sin$\varphi$ but I just did not get this answer... I don't think there is anything wrong with my computation of the determinant.This is...Read more

I can understand this through various examples found in the internet but I can't quite intuitively understand why the determinant of the derivative(in its most general form)-the Jacobian-gives the change of volume factor that arises when we change variables in, say, an integral. I mean, why does the determinant of the matrix consisting of the derivatives of the original variables wrt the new variables give a number that corresponds to how much the infinitesimal volume has changed? How can we geometrically connect the derivatives that are the ...Read more

I have a series of questions, in various degrees of befuddled muddledness (and they are related to my previous questions: this and this)First question: how do I do a change of variable if the determinant of the jacobian is singular?The setting for this question is as follows: I have one $n$-dimensional standard gaussian random variable $u \sim N(0,I)$ and a fixed $v \in \mathbb{R}^n$. Then I define the random variable$$z = u - \frac{u^Tv}{v^Tv}v$$and I'd like to derive a density for $z$. So the Jacobian:$$\frac{dz}{du} = I - \frac{vv^T}{v^Tv}$$...Read more

$$ \int \int_S (x^2+y^2)d\sigma$$Where S is the unit sphere centered at (0,0,0), and $\sigma$ is surface area.I arrived at the correct answer of $\large \frac{8\pi}{3}$, but I took an (educated?) guess on the set-up of the integral.Instead of the usual volume element of the sphere, in spherical coordinates,$$dV=\rho^2sin(\phi)d\rho d\theta d\phi$$I just simply omitted the $\rho^2 d\rho$ factor of the Jacobian and guessed that the remaining factor $$sin(\phi) d\theta d\phi$$represents the "surface area element" of the sphere. Integration gave me...Read more

Use a triple integral to find the volume of the solid: The solid enclosed by the cylinder $$x^2+y^2=9$$ and the planes $$y+z=5$$ and $$z=1$$This is how I started solving the problem, but the way I was solving it lead me to 0, which is incorrect. $$\int_{-3}^3\int_{-\sqrt{9-y^2}}^{\sqrt{9-y^2}}\int_{1}^{5-y}dzdxdy=\int_{-3}^3\int_{-\sqrt{9-y^2}}^{\sqrt{9-y^2}}\left(4-y\right)dxdy=\int_{-3}^3\left[4x-xy\right]_{-\sqrt{9-y^2}}^\sqrt{9-y^2}dy= {8\int_{-3}^3{\sqrt{9-y^2}}dy}-2\int_{-3}^3y{\sqrt{9-y^2}}dy$$If this is wrong, then that would explain wh...Read more

How do I use Cylindrical Coordinates to find volume of a solid in the first Octant that is bounded by the cylinder $x^2 + y^2 = 2y$, the half cone $z = \sqrt{x^2 + y^2}$, and the $xy$-plane.I have drawn the region of integration and obtained this:$\int_0^2 \int_0^\sqrt{2y-y^2}\int_0^\sqrt{x^2 + y^2} dzdxdy$Is this correct and from here were do I apply the cylindrical coordinates?...Read more

my question is can cylindrical coordinates be somehow changed in order? I mean the standard cylindrical coordinates suppose a line of length $r$ that makes an angle $\theta$ with the $x$ axis and let $z$ vary and then $x=rcos\theta$ and $y=rsin\theta$ and the line $r$ (at least initially) is in the $xy$ planenow can we suppose the the line of length $r$ makes an angle with the $x$ axis but instead is in the $xz$ plane and let $y$ vary and then $x=rcos\theta$ and $z=rsin\theta$ and if this can be done then what is wrong with this?the problem is ...Read more