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).

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3
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0answers
152 views

integral of product of curve with itself in three dimensional

I have the next problem: Let $\gamma:[0,1]\to R^3$ differentiable curve piecewise, and let $\Delta_r=\{(s,t)\in [0,1]^2| |\gamma(s)-\gamma(t)|<r\}$, i want to know if: ...
2
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0answers
21 views

Simplified Helmholtz decomposition

Suppose we are given a vector field $$\vec{a}= (x+y+z)\vec{i}+y\vec{j}+z\vec{k} \tag{$*$}$$ And we want to write it as a linear combination of a potential field and a solenoidal field, meaning ...
1
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2answers
33 views

Vector Identity Question

I am having some trouble with this question regarding vector diffiriential operators. It seems easy and I am not sure what I am missing. The question: Prove: $$ ...
0
votes
1answer
36 views

Computing the Jacobian determinant for a change of variables,

Why do we compute the partial derivatives in terms on the new variables and not with respect to the old variables? Can someone say this intuitively, so that I have the best chance of remembering why? ...
1
vote
1answer
64 views

Formula for the gradient of $F(\rho,\phi,z)$

Suppose $F(\rho,\phi,z)$ is continuously differentiable, I am interested in showing that the maximum directional derivative of $F$ at any point is given by ...
2
votes
0answers
41 views

Stuck while trying to simplifying this PDE identity

This is related to a fluids HW: Let $u(x)$ be incompressible vector field on n-D space, $x\in \Omega\subset R^n$ and let $\theta(x)$ be a scalar field s.t. $\int_{\Omega} \theta(x)dx=0$, $\nabla.u=0$ ...
1
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2answers
43 views

Quickest way to determine if a vector field is conservative?

Say I have some vector field given by $$\vec{F} (x,y,z)=(zy+\sin x)\hat \imath+(zx-2y)\hat\jmath+(yx-z)\hat k$$ and I need to verify that $\vec F$ is a conservative vector field. What would be the ...
3
votes
0answers
21 views

Any unit speed reparametrization $\beta=\alpha(h)$ of $\alpha$ is reparametrized by an arc length function.

These are definitions given from Barret O'neill's Elementary Differential Geometry. Definition $1$. A curve in $\mathbb{R}^3$ is a differentiable function $\alpha: I\to \mathbb{R}^3$ from an open ...
0
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0answers
25 views

Prove the following identitie

Given the vector fields F and G in $R^3$, I have learnt the grad(vector of derivatives) or del and curl(cross product) of a function. But I get stack when trying to prove the following identities ...
0
votes
1answer
18 views

What is nabla scalar (a.u) where a is a scalar field and u a vector field?

We have a domain D of say R² and a function a from D to R and a function u from R² to R² what is Nabla dot (au) ? If u were from R² to R we could have simply used the product rule
0
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0answers
24 views

The proof by partial derivatives and vector calculus

Prove that if $f(x,y,z)$ is a composite function $F(u)$, where $u=g(x,y,z)$. I am trying to show that $\nabla f=F'(u)\nabla g$. I have learnt the vector calculus and vector fields but the composite ...
1
vote
1answer
21 views

vector triple product does not go from $[\dot{M} \times [H \times M]]+[M \times [H \times \dot{M}]]$ to $2\dot{M}(H,M)$

I have the following term $ [M \times [H \times M]] $ under the time derivative. After using the rule of the derivate of the product I get: $$[\dot{M} \times [H \times M]]+[M \times [\dot{H} \times ...
1
vote
2answers
24 views

Computing partial derivatives using three implicitly defined equations

The three equations $x^2-y\operatorname{cos}(uv)+z^2=0$ $x^2+y^2-\operatorname{sin}(uv)+2z^2=0$ $xy-\operatorname{sin}u\operatorname{cosv}+z=0$ define $x,y,z$ as functions of $u,v$. Compute the ...
1
vote
2answers
43 views

Find a unit tangent vector to a curve that is an intersection of two surfaces.

The intersection of the two surfaces given by the Cartesian equations $2x^2+3y^2-z^2=25$ and $x^2+y^2=z^2$ contains a curve $C$ passing through the point $P=(\sqrt{7},3,4)$. These equations may be ...
0
votes
1answer
30 views

Showing that $\nabla (\alpha f) = \alpha \nabla f$ for constant $\alpha$

I want show that del of alpha times a vector function for is equal to alpha times del of fun using. Alphar is a constant hence it should be factories out after finding partial derivetives,but how do ...
0
votes
1answer
28 views

How to determine the maximum rate of increase in temperature

Suppose that the temperature at a point $(x,y,z)$ in space is given by $T(x,y,z)=\frac{80}{1+x^2+2y^2+3z^2}$ where $T$ is measured in degrees celsius and $x$,$y$ and $z$ in meters. In which ...
1
vote
1answer
20 views

Explicitly demonstrating Stokes' theorem over a tetrahedron.

Consider the vector field: $$\vec{F} = \left(2x-y,-yz^2,-y^2z\right)$$ We are to explicitly show stokes theorem: $$\oint_\Gamma \vec{F}\cdot d\vec{x} = \int\int_{\partial V}\nabla \times \vec{F}\cdot ...
1
vote
1answer
22 views

Showing that a function is the gradient of another function

How do I show that this function; $ f = \frac{\vec{r}-\vec{X}t}{|\vec{r}-\vec{X}t|^3}$ $\vec{X} = (x_1,x_2,x_3)$ and $\vec{r} = (x,y,z)$ is the gradient of another function? like so: $ f = \nabla ...
4
votes
2answers
19 views

Problems on orthogonality and tangency in 3-space.

Find the set of all points $(a,b,c)$ in 3-space for which the two spheres $(x-a)^2+(y-b)^2+(z-c)^2=1$ and $x^2+y^2+z^2=1$ intersect orthogonally. A cylinder whose equation is $y=f(x)$ is tangent to ...
2
votes
1answer
21 views

Let $\mathbf{r}=(x,y,z)$,$r=||\mathbf{r}||$. Show the following equation on $B\cdot \nabla (A\cdot \nabla (\frac{1}{r}))$

Let $\mathbf{r}=(x,y,z)$ and let $r=||\mathbf{r}||$. If $A$ and $B$ are constant vectors show that: $$B\cdot \left(\nabla \left (A\cdot \nabla \left(\frac{1}{r}\right)\right)\right)=\frac{3A\cdot ...
1
vote
3answers
43 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
13 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
0answers
16 views

Find a pair of linear Cartesian equations for the line which is tangent to both the surfaces $x^2+y^2+2z^2=4$ and $z=e^{x-y}$ at the point $(1,1,1)$.

Find a pair of linear Cartesian equations for the line which is tangent to both the surfaces $x^2+y^2+2z^2=4$ and $z=e^{x-y}$ at the point $(1,1,1)$. I think the line that is tangent to both the ...
0
votes
1answer
19 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
35 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
33 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
58 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
29 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
28 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
117 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
63 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
48 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
49 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
44 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
21 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
32 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
11 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
61 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
19 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
29 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
51 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
90 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
6 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 ...