Questions related to understanding line integrals, vector fields, surface integrals, the theorems of Gauss, Green and Stokes. Some related tags are (multivariable-calculus) and (differential-geometry).

learn more… | top users | synonyms

1
vote
3answers
46 views

If $f(x,y,z)=(r \times A)\cdot (r\times B)$, where $r=(x,y,z)$, show that $\nabla f(x,y,z)=B\times (r\times A)+A\times (r\times B)$.

If $f(x,y,z)=(r \times A)\cdot (r\times B)$, where $r=(x,y,z)$ and $A$ and $B$ are constant vectors, show that $\nabla f(x,y,z)=B\times (r\times A)+A\times (r\times B)$. I'm a bit lost on this ...
0
votes
1answer
14 views

If $||\nabla f(x,y)||^2=2$, determine constants $a$ and $b$ such that $a(\frac{\partial g}{\partial u})^2-b(\frac{\partial g}{\partial v})^2=u^2+v^2.$

The change of variables $x=uv$, $y=\frac{1}{2}\left(u^2-v^2\right)$ transforms $f(x,y)$ to $g(u,v).$ If $\left\|\nabla f(x,y)\right\|^2=2$ for all $x$ and $y$, determine constants $a$ and $b$ such ...
2
votes
1answer
28 views

Concerning an application of the divergence theorem

I was studying the derivation of Helmholtz decomposition through Wikipedia and I've come across an application of the divergence theorem which I'm not familiar with. I'd appreaciate if you could help ...
2
votes
1answer
25 views

show that the equation $r_1+r_2= \text{constant}$ implies the relation $\mathbf{T}\cdot \nabla(r_1+r_2)=0,$

This is a problem from Apostol's Calculus, which I have difficulty solving. If $r_1$ and $r_2$ denote the distances from a point $(x,y)$ on an ellipse to its foci, show that the equation $r_1+r_2= ...
0
votes
1answer
22 views

Find a vector $V(x,y,z)$ normal to the surface $z=\sqrt{x^2+y^2}+(x^2+y^2)^{3/2}$

(a) Find a vector $V(x,y,z)$ normal to the surface $$z=\sqrt{x^2+y^2}+(x^2+y^2)^{3/2}$$ at a general point $(x,y,z)$ of the surface, $(x,y,z)\neq (0,0,0)$. (b) Find the cosine of the angle $\theta$ ...
1
vote
3answers
46 views

Evaluate the directional derivative of $f$ for the points and directions specified

Evaluate the directional derivative of $f$ for the points and directions specified: (a) $f(x,y,z)=3x-5y+2z$ at $(2,2,1)$ in the direction of the outward normal to the sphere $x^2+y^2+z^2=9.$ (b) ...
2
votes
3answers
36 views

Potential for integration

I have the following function inside an integral $$\frac{2xdx + 2ydy + 2zdz}{x^2 + y^2 + z^2}$$ I need to find the potential for solving the integral, but I don't know how to transform it into a ...
2
votes
0answers
47 views

Area of a GREEN-region

a) Show that the area of GREEN-region B (which is defined by: $A\subseteq R^2$, $A=B_1\cup ...\cup B_m$, and $\mathring{B}_i\cap \mathring{B}_j=\emptyset$) in the plane is defined by: ...
3
votes
1answer
70 views

Find the flux across a part of the surface $\frac{x^2}{4}+\frac{y^2}{9}+\frac{z^2}{16}=1$

Consider the vector field $$F(x, y, z)= \frac{(x{\rm i} + y{\rm j} + z{\rm k})} {(x ^2+ y ^2 + z ^2)^\frac{3}{2}},$$ and let $S$ be the part of the surface ...
0
votes
1answer
34 views

The area of surface obtained by rotating a rectifiable curve

Let $\Gamma :X=\gamma(t),a\leq t\leq b$ be a rectifiable parameterized curve in the $(x,z)$-plane of $R^3$, which means $\gamma:[a,b]\to R^3$ is a $C^1$-mapping with $\gamma(t)=(x(t),0,z(t))^T$ and ...
2
votes
0answers
59 views

Directional derivative expression

We have $n=\sqrt{{\mathbf N}\cdot{\mathbf N}}$, where ${\mathbf N}$ is the normal vector to a curve, let's accept ${\mathbf N}=\ddot{{\mathbf r}}$, say the curve is unit-speed. We also have a scalar ...
1
vote
2answers
31 views

For $x \, \epsilon \,R^n$, prove $\int_{|x| \leq 1} x_i x_j dx = 0$

Let $x \, \epsilon \, R^n$, with $x_i$ corresponding coordinates. Then $|x| \leq 1$ is the unit ball in $R^n$. How can I easily prove that $\int_{|x| \leq 1} x_i x_j dx = 0$ if $i \ne j$? It is kind ...
0
votes
1answer
32 views

Find directions in which f increases and decreases the most rapidly. Then find the derivatives of f at these directions. [closed]

$f(x,y)=3x^2+2xy+4y^2$ This is what I have to find: direction of fastest increase I found the gradient vector: $f(x,y)=(6x+2y)i+(2x+8y)j$ $f(9,2) = \langle 58,34\rangle$ Is this the direction of ...
5
votes
1answer
124 views

Divergence theorem for a second order tensor

I want to integrate by part the following integral in cylindrical coordinates $$\int \vec{r} \times (\nabla \cdot \overline{T}) ~d^3\vec{r} $$ where $\overline{T}$ is a second order symmetric tensor ...
1
vote
3answers
110 views

Derivation of divergence in spherical coordinates from the divergence theorem

I'm trying to find the expression of the divergence of a vector field $\vec{E}$ in spherical coordinates from the theorem : $$\iint_{S(V)}(\vec{E}.\vec{n})dS = \iiint_{V}div(\vec{E})dV$$ but if I ...
1
vote
0answers
21 views

Notation for vector valued integration

Suppose we have a vector field $\mathbb v=(v^1,v^2)$ on the two dimensional torus $\mathbb T^2$ and we wish to compute $\int_{\mathbb T^2} \mathbb v \cdot \Delta \mathbb v $, where the Laplacian acts ...
0
votes
1answer
54 views

Intrinsic definition of divergence and curl

Are the intrinsic definitions of divergence and curl the theorems of Green-Ostrogradski and Stokes-Ampere respectively ? What is a rigorous derivation of their expression in a coordinate system ?
2
votes
4answers
51 views

$\vec{a}\times(\vec{a}\times\vec{R})-\vec{b}\times(\vec{b}\times\vec{R})$

I have $\vec{a}\times(\vec{a}\times\vec{R})-\vec{b}\times(\vec{b}\times\vec{R})$, my textbook says that this equals $((\vec{a}\times\vec{a})-(\vec{b}\times\vec{b}))\times\vec{R}=-(a^2-b^2)\vec{R}$. I ...
0
votes
1answer
50 views

Cross product of the gradient of two functions

I am having a bit of a confusion with some claims I keep finding on a book of Fluid Dynamics. Let's say we have two functions in 3-D space, $f(\mathbf{x})$ and $g(\mathbf{x})$, with the following ...
0
votes
1answer
24 views

How should one define the cross product for two vector valued functions?

In many textbooks regarding vector analysis, the cross product is mention in the section about the properties of the derivative of vector functions, but there seems to be no explanation what that ...
0
votes
1answer
36 views

Gradient and Laplacian in integral.

Let $u,v,f$ be functions of $\mathbb{R}^n$ to $\mathbb{R}$, with compact support in a domain $U$, this formula $$\int_{U} f(x) (Du \cdot Dv) dx = \int_{U} f(x)(u D(Dv)) dx = \int_{U} f(x) u(x) \Delta ...
0
votes
0answers
12 views

Applying different boundary conditions in a quasi Helmholtz problem.

I have tried to solve this exercise from Applied Partial Differential Equations-Richard Haberman, Consider the two-dimensional eigenvalue problem $$ \nabla^2\phi+\lambda\sigma(x,y)\phi=0 $$ ...
1
vote
1answer
67 views

Vector calculus identity for $\nabla\times(\vec{b}\cdot\nabla)\vec{b}$

I'm going through a paper on turbulence and in it the author uses the following $$ ...
0
votes
0answers
24 views

Proving Euler's theorem for homogeneous functions.

This problem is from Apostol's Mathematical Analysis. Let $f$ be defined on an open set $S$ in $R^n$. We say that $f$ is homogeneous of degree $p$ over $S$ if $f(\lambda x)=\lambda ^p f(x)$ for every ...
0
votes
1answer
30 views

Does a fluid with $0$ divergence have $0$ density?

I'm starting a course on Vector Calculus, and I got to the intuitive relation between the divergence and the density of the fluid, namely that we can see the divergence as the opposite of the change ...
1
vote
2answers
53 views

proof of laplacian $1/\rho$ in cylindrical coordinates at $\rho = 0$

In spherical coordinates, it can be proved that the laplacian of $r = \sqrt{x^2+y^2+z^2}$ at the origin is \begin{align} \nabla^2 \dfrac{1}{r} = -4\pi\delta(\vec{r}) \end{align} as demonstrated in ...
2
votes
1answer
15 views

Given a function $f$ defined in $R^2$. Let $F(r,\theta)=f(r\cos\theta,r\sin\theta).$ Verify a formula of the modulus of the gradient.

Given a function $f$ defined in $R^2$. Let $$F(r,\theta)=f(r\operatorname{cos}\theta,r\operatorname{sin}\theta).$$ Verify the formula $$|\nabla f(r\operatorname{cos}\theta, ...
7
votes
0answers
93 views

Very difficult surface integral

Compute the surface integral: $$\int_S({x\over \sqrt{x^2+y^2+z^2}}, {y\over \sqrt{ x^2+y^2+z^2}}, {z\over \sqrt{x^2+y^2+z^2}}), \cdot \vec n \ dS$$ where $S: x^3+y^3+z^3=a^3$ The first ...
1
vote
0answers
7 views

Conservative field $F$ on not simple connected set

Give an example of a field $F:D\subseteq \mathbb R^2 \to \mathbb R^2$ such that $D$ is a doubly connected set (that is $D$ has on "hole") but $F$ is conservative And if $D$ is a triply connected ...
0
votes
1answer
51 views

laplacian of $1/\rho$ in cylindrical coordinates

In spherical coordinates, I believe that the laplacian of $1/r$ is zero everywhere except at $r = 0$ or \begin{align} \nabla^2 \dfrac{1}{r} = -4\pi \delta^{(3)}({\vec{r}}). \end{align} where $r$ is ...
3
votes
2answers
29 views

applying the product rule to a vector analysis question

I have been doing doing this problem $∇ × (\varphi∇\varphi)=0$ I am just having trouble applying the product result i get which is below. $$i(( \frac {d}{dy} )(\varphi \frac {d}{dz} \varphi) - ...
0
votes
1answer
28 views

An application of Greens's theorem

Apply Green's theorem to prove that, if $V$ and $V'$ be solutions of Laplace's equation such that $V=V'$ at all points of the closed surface $S$, then $V=V'$ throughout the interior of $S$. ...
3
votes
2answers
56 views

Computing line integral using Stokes´theorem

Use Stokes´ theorem to show that $$\int_C ydx+zdy+xdz=\pi a^2\sqrt{3}$$ where $C$ is the curve of intersection of the sphere $x^2+y^2+z^2=a^2$ and the plane $x+y+z=0$ My attempt: By Stokes´ theorem ...
0
votes
0answers
20 views

An electron in a TV tube is beamed horizontally at a speed of 4.3 * 10^6 meters per second

An electron in a TV tube is beamed horizontally at a speed of 4.3 * 0^6 meters per second toward the face of the tube 31 cm away. How far will the electron drop before it hits? (Assume ideal ...
1
vote
2answers
31 views

How to prove that $(\vec n\cdot\nabla)\vec n$ is orthogonal to $\vec n$ for unit vector $\vec n$?

I'm trying to prove or disprove that if $\vec n(x,y,z)$ is a unit vector, then $(\vec n\cdot\nabla)\vec n$ is orthogonal to $\vec n$. For this I first tried to compute $\vec n\cdot((\vec ...
0
votes
1answer
67 views

$\nabla \times F=0$ implies that $F$ is conservative

Prove that if $F:\mathbb R^3\to \mathbb R^3$ is a vector field so that $\nabla\times F=0$ $\forall x\in \Omega\subset \mathbb R^3$ (where $\Omega$ is an open simply connected set), then $F$ is a ...
0
votes
0answers
26 views

Show that $\iint_S (n\times \nabla)f\, dS=\int_C tf\, ds$ and $\iint_S (n\times \nabla)\times f \,dS=\int_C t\times F\, ds$

If $S$ be a closed region lying on a surface and bounded by the curve $C$ and $n$ be the unit positive normal vector to $S$, and $f$ and $F$ be two fields with continuous fiest derivatives in $S$, ...
3
votes
2answers
72 views

Prove that $\int_S n\times r dS=0$

If $r$ be the position vector of a point on a closed surface $S$ and $n$ be the unit normal (outward) vector to $S$, then prove that $$\int_S n\times r\,dS=0$$ Attempt: $r=xi+yj+zk$, ...
0
votes
2answers
38 views

What does $\text{div} (A \text{ grad }b)$ mean?

I often see this term in my Applied Mathematics course. If $\text{grad }b= \frac{\partial b}{\partial x}\hat{i}+\frac{\partial b}{\partial y}\hat{j}+\frac{\partial b}{\partial z}\hat{k}$, then would ...
0
votes
0answers
15 views

How to find the value of standard coordinate frame in a new coordinate frame?

I have a custom coordinate frame which has T as a point and A, B, C are three orthogonally normalized vectors whose coordinates are T = [Xt Yt Zt], A = [Xa Ya Za], B = [Xb, Yb, Zb] and C = ...
1
vote
1answer
29 views

How to simplify this expression using tensor notaion?

$\nabla^2 (\phi A)-A \nabla^2 \phi -2(\nabla \phi \cdot\nabla)A$ Where $A,\phi$ are any sufficiently smooth vector and scalar fields respectively.
0
votes
0answers
38 views

a problem on stokes' theorem

the problem is as following Use stokes theorem to evaluate $\oint F.dr$ where, F = (-2Z) i + (X) j - (X) k , C is the ellipse $X^2 + Y^2 = 1 $ and $ Z = Y + 1 $ my solution is to get $curl F $ ...
0
votes
1answer
19 views

Calculating a line integral around a closed curve.

Let $u_0$ be a fixed vector, and let $b=u_0\times r$, where $r$ is the position vector $x\hat{i}+y\hat{j}+z\hat{k}$. What is $\int_C b.\hat{T}ds$, where $C$ is a closed curve? Assuming ...
0
votes
0answers
36 views

The gradient of the magnitude of the cross product of a constant vector and the position vector

If $ \underline c$ is a constant vector and $\underline r$ is the position vector. How can I show that $ \lvert \underline c \land \underline r \rvert grad \lvert \underline c \land \underline r ...
1
vote
2answers
16 views

Prove that, for $n, l \in \mathbb{N}$ the identity $\vec\nabla \times (f^n \vec\nabla(f^l)) = \textbf{ $\vec 0$} $

a) Let $f$ and $g$ be two smooth scalar fields. Prove the following identity: \begin{equation} \vec\nabla \times (f \vec\nabla g) + \vec\nabla \times (g \vec\nabla f) = \textbf{$\vec 0$} ...
1
vote
1answer
45 views

Intuitive explanation of the potential function of a vector field

Suppose I have some vector field $$\vec{f}(x,y)=\begin{pmatrix}A(x,y)\\B(x,y)\end{pmatrix}$$ then the potential function (if the field is conservative) can be found by integrating $A$ with respect ...
1
vote
1answer
36 views

Surface Integral over a Vector Field question

pretty basic question but I can't seem to work it out: Question: Let S be the triangle with vertices $\mathbf{a}=(1,2,3),\mathbf{b}=(1,1,1),\mathbf{c}=(3,1,2)$ with unit normal chosen such that the ...
2
votes
2answers
54 views

Linearity of Multilinear Maps

If $f: \mathbb{R^n} \rightarrow \mathbb{R^m}$, with $n>1$, is a multilinear map, is $f$ linear? I think $f$ is only linear for the special case that the range of $f$ consists of a single element, ...
2
votes
1answer
16 views

Orthogonal decompostion for $u^´(t)$

$u(t) $ is differentiable vector function in $\mathbb{R}^3$ on $[a,b]$ and $u(t) \neq 0$ for all t. $u^´(t)$ is the derivative of $u(t)$ and is orthogonal for $t \in (a,b)$ for all t $\implies$ ...
1
vote
3answers
89 views

Can you multiply two vectors $v$ and $u$ using multiplication instead of dot or cross products

On Wikipedia page on vector multiplication, they gave several ways to "multiply" two vectors, the most well known being the dot product and the cross product I am curious whether we can multiply two ...