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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2
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3answers
55 views

what does $(A\cdot\nabla)B$ mean?

I was studying a physics book and I saw this expression $$(A\cdot\nabla)B$$ where $A$ and $B$ are vectors. What's the definition of this? I've also seen this in some identities
0
votes
0answers
18 views

Gradient of 3d delta-function

I need to evaluate the following expression: $\int \mathrm{d}\boldsymbol{r} \left[\nabla_{\boldsymbol{R}_\alpha}\delta(\boldsymbol{r}-\boldsymbol{R}_\alpha)\right]v(\boldsymbol{r})$ and I want to ...
0
votes
0answers
9 views

Nonlinear Conjugate gradient for vector valued multi variable functions?

So far what I have found in online and in Numerical recipes book describe algorithm for scalar value multi variable function. Can anyone point me to the algorithm for nonlinear conjugate gradient for ...
0
votes
1answer
16 views

Potential of vector field is undefined on Y-axis although field is defined

I'm having the following vector field: $$\vec{F}(x,y) = (\frac{-y}{x^2+y^2},\frac{x}{x^2+y^2})$$ The field is conservative in $\mathbb{R}^2 \backslash (0,0)$ as long as your curve doesn't encircle $(...
0
votes
0answers
23 views

Confusion with chain rule when proving statement about tangent plane to a point in a manifold

I'm trying to prove the following: If $f:\mathbb{R}^3 \to \mathbb{R}$ is a differentiable function, $a \in \mathbb{R}$ is a regular value of $f$ and $S=f^{-1}(a)$, then for all $p \in S$ the tangent ...
1
vote
1answer
33 views

curl and stokes application

I cannot fin the flux of $$F(x,y,z)=(y^2cos(xz),x^3e^{yz},-e^{-xyz})$$ through the portion of sphere $$\Sigma = \{x^2+y^2+(z-2)^2=8, z\ge0 \}$$ I think Stokes th. must be used, so in spherical ...
1
vote
1answer
42 views

Calc 3: Calculate Work Done on Particle [on hold]

I've been working on this problem for a while and I'm pretty stuck. I tried it multiple different ways, by the last time I attempted it I realized that I hadn't converted kilometers to meters the ...
0
votes
0answers
17 views

Let $\phi:(0,1)\to \mathcal{L}(\mathbb{R}^m,\mathbb{R}^n)$ given by $\phi(t)=A(tx)$, then $\phi'(t). h=(A'(tx). x). h$ or $(A'(tx). h). x$?

Let $U$ be an open ball centered in $0$ in $\mathbb{R}^m$. Given $\phi:(0,1)\to \mathcal{L}(\mathbb{R}^m,\mathbb{R}^n)$ be defined by $\phi(t)=A(tx),$ where $A:U\to \mathbb{R}^n$ and $x\in U$, which ...
0
votes
2answers
30 views

Solving vector equation 3

Solve for $\bar{x}$ and $\bar{y}$ $$\bar{x}+\bar{y}=\bar{a},~~ \bar{x}\times \bar{y}=\bar{b},~~ \bar{x}.\bar{a}=1$$ Attempt: $\bar{x}+\bar{y}=\bar{a}$ dot by $\bar{a}$, we get $1+\bar{a}.\bar{y}=|...
0
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1answer
29 views

Solving vector equation 2

Using vector method, show that the vector equation $$\bar{x}\times \bar{a}+(\bar{x}.\bar{b})\bar{c}=\bar{d}$$ is satisfied if $$\bar{x}=\lambda \bar{a}+\bar{a}\times \frac{\bar{a}\times (\bar{d}\...
-1
votes
1answer
53 views

Solving vector equation 1 [on hold]

Using vector method solve $p \bar{x}+\bar{x}(\bar{x}.\bar{b})=\bar{a}\times \bar{b}+\bar{c}$ How to solve $\bar{x}$ from such vector equation. Please help.
1
vote
1answer
49 views

Volume of solid lies under $z=x^2+y^2$ [on hold]

Find the volume of solid lies under $z=x^2+y^2$ above $x$-$y$ plane and inside the cylinder $x^2+y^2=2x$. I know, for volume we have to us $V=\iiint { \mathrm dx\mathrm dy\mathrm dz}$ but i was not ...
0
votes
0answers
9 views

Gamma in 3D coordinate system [on hold]

Gamma along z axis apparently look clockwise but it is measured along anti-clockwise direction. What's the reason behind this?
0
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0answers
34 views

How calculate intersection directly without Stokes' theorem?

Calculate the line integral directly without Stokes' theorem: \begin{gather*} \oint_\gamma \mathbf{F} \cdot d\mathbf{r} \end{gather*} \begin{gather*} \mathbf{F}(x,y,z)=(2z-3y) {\hat{\mathbf{i}}} + (3x-...
0
votes
1answer
20 views

Calculate the gradient of a function that is written with abstract vectors

:) I am supposed to calculate the gradient of the following function: $$f(\mathbf{w})=\sum^{n}_{i=0}\log(1+\exp(-y_i\mathbf{w}^T\mathbf{x}_i))+\frac{1}{b}\sum^{n}_{i=0}w_i^4$$ Where $\mathbf{x} \...
1
vote
0answers
21 views

Transformation of the gradient

For a function $f\in C^2$, $f:\mathbb{R}^n\to\mathbb{R}$ and a point $x\in\mathbb{R}^n$ with $\nabla^2f(x)$ positive definit one can calculate the new point $x^+=x+s$ as follows: Change the ...
0
votes
0answers
36 views

Calculus & Analytic Geometry VS Vector Calculus

This question may be applicable for Academia SE, however this is strictly math-oriented and requires math whizzes' opinions. I intend to go to a tech institute to get a BS majoring in Computer ...
3
votes
1answer
52 views

Proving an integral relation (isotropic function)

In Hansen-McDonald's book Theory of Simple Liquids the following relation is often used: We want to evaluate the integral $$\int_V f(\vec r_1, \vec r_2) d \vec r_1$$ We observe that if the function ...
3
votes
1answer
48 views

How is “expressing” a differential operator “in cylindrical coordinates” rigorously defined?

I'm a mathematician (with little knowledge of differential geometry) trying to study physics. One of the greatest problems is the language regarding coordinate transformations. I tend to think of such ...
0
votes
1answer
39 views

Application of Vector Calculus;Line integrals [closed]

A force is given by $\ F=(cxy \ i + x^6y^2\ j)$, where $i$ and $j$ are unit vectors.The force acts on a particle which must move from (0,0) to the line x=1 along the curve $y=a(x^b)$ where $a>0,b&...
1
vote
1answer
16 views

Gradient of function, which has codomain R^2 or bigger.

For example I have a function: $$f(x_1,x_2) = \begin{bmatrix} x_1x_2^2 + x_1^3x_2\\x_1^2x_2 + x_1 + x_2^3\\\end{bmatrix}$$ Is it possible to find a gradient of this function? Because knowing the ...
4
votes
2answers
75 views

the uniform convergence of the sequence of functions

Let $f_1:[a,b]\rightarrow \mathbb{R}$ be a Riemann integrable function. Define the sequence of functions $f_n:[a,b] \rightarrow \mathbb{R}$ by $f_{n+1}(x)=\int_a^x f_n(t)dt,$ for each $n\ge 1$ and ...
0
votes
0answers
16 views

How is this surface integral changed into a volume integral?

A solution $V(\mathbf{x},t) \in C^2(\mathbb{R}^3 \times \mathbb{R}_+)$ to a certain linear hyperbolic partial differential equation can be expressed as: $$V({\mathbf x}, t)= \frac{1}{4 \pi}\int_0^t\...
-1
votes
0answers
14 views

Is there any equality for the integral of the product of normal derivative?

I am trying to get the proof of $\int\int_DD_uf(x) D_ug(x) dx$. For example in Green Theorem, in integral we use the product of $ \nabla$, when it comes to normal derivative, how can I organize the ...
0
votes
1answer
43 views

Flux of a vector field $F(x,y,z)$

I have a circle in the $yz$-plane centered at $(0,2,0)$ with radius $1$. The surface $\Sigma$ is obtained by rotating the circle around the $z$-axis. I want calculate the flux of the vector field $...
0
votes
0answers
53 views

Stokes theorem for the flux through a surface

I have a surface $\phi(u,v)=(u\cos v,1-u,u\sin v) ; u\in[0,1]; v\in[0,2\pi]$. I want calculate the flux through $\phi$ of $F(x,y,z)=(z+\arctan y,\frac{x^5}{1+z^2},x^2ze^{y^2}) $ as $\int_{\phi} <...
1
vote
0answers
37 views

Green's formulae, Stokes theorem, Gauss theorem, divergence theorem and Gauss-Green theorem?

I am getting really confused about the Green's formulae, the Divergence theorem and all those related equalities. For example, How is this formula exactly called? $$\int_\Omega \frac{\partial u}{\...
1
vote
1answer
25 views

Derivative of Lattice Laplacian

The lattice Laplacian is defined as, $$ \nabla_L^2x_j \equiv \frac{x_{j+1} - 2x_j + x_{j-1}}{a^2} $$ where the lattice spacing, $a$, is a constant. The derivative, with respect to $x_i$, then gives, ...
0
votes
0answers
61 views

Proof of an equation in vector calculus

How do I prove the following by integration by parts: $$\iiint \left( \vec{\triangledown} . \dfrac{\vec{I}}{r} \right)dV=\iint\dfrac{1}{r} \vec{I}.\hat{n} dS-\iiint\dfrac{1}{r} \left( \vec{\...
3
votes
2answers
42 views

“Inverse” Helmholtz Decomposition

So I am trying to write a report on the Helmholtz decomposition theorem on $\mathbb{R}^3$. The theorem states that under certain conditions, every vector field $\textbf{F}:U \subseteq \mathbb{R}^3 \to ...
1
vote
1answer
12 views

Finding maximum and minimum values of the rate of change of parametric graph

Consider the ellipse $ r'(t) = \langle3\cos(t),4\sin(t)\rangle$ for $ 0\le t \le 2. $ (a) At what points $\|r'\|$ have maximum and minimum values? (b) At what points does the curvature have ...
1
vote
0answers
42 views

Suggestion of books on Integral Calculus of Several Variables

I'd like recommendations of books on integral calculus of several variables (double integral until Gauss's theorem) that contains challenging(hard) problems. And I'd like books in languages other than ...
1
vote
0answers
10 views

line/surface integral parameterization questions

I have the following question: Let $i, j, k$ denote the unit vectors on $x, y, z$ axes of cartesian coordinates, respectively. Calculate the line integral $\int_c A \cdot d\textbf{r} $ and the ...
0
votes
1answer
15 views

Helmholtz Decomposition on $\mathbb{R}^3$ Proof

I am trying to prove the Helmholtz decomposition theorem which states that given a smooth vector field $\mathbf{F}$, there are a scalar field $\phi$ and a vector field $\mathbf{G}$ such that \begin{...
2
votes
2answers
56 views

$\nabla \cdot (\nabla \times \vec A) = 0 $ Proof

Show that $\nabla \cdot (\nabla \times \vec A) = 0 $ for an arbitrary differentiable vector field $\vec A$ using Stokes' Theorem for an arbitrary closed surface S followed by Gauss' Theorem. An ...
0
votes
2answers
64 views

Calculating a surface integral over some ellipsoid

a) Let $S$ the surface $4x^2+9y^2+36z^2=36$, $z \ge 0$. Let $\vec{F}=y\vec{i}+x^2\vec{j}+(x^2+y^4)^{3/2}\sin(e^{xyz})\vec{k}.$ Calculate the integral $\iint (\text{curl }\vec{F})\cdot\vec{n}\,dS$. b)...
3
votes
0answers
48 views

Surface Integral of a Vector Field

Consider the vector field $\vec{F}(x,y,z)=(x,y,z)$, and the surface parametrized by $\Phi(u,v)=(uv,\frac{u^{2}+v^{2}}{2},\frac{u^{2}-v^{2}}{2})$ where $0\leq u\leq 2 $ and $0\leq v \leq 4$. Evaluate ...
0
votes
0answers
26 views

Applying Green's identity for solving an equation of surface integrals

Prove the following: Let $R = \text{cl}(\text{U}), \text{U} \in \mathbb{R}^{3} $ an open set limited by a closed surface S oriented by exterior unitary normal $\vec{n}$ and $u:R \rightarrow \mathbb{R}...
0
votes
0answers
14 views

Maxwell-type system: how to solve?

$\DeclareMathOperator{\curl}{curl}\renewcommand{\div}{\mathop{\mathrm{div}}}$ Consider the following system of equations for the unknown vector field $A$ in the unit 3d ball $B$ with boundary $S$: $$ ...
0
votes
0answers
38 views

Find $\iint (\nabla \times F)\cdot dS$ if S the surface of the sphere $x^2+y^2+z^2=a^2$

Find $\iint (\nabla \times F)\cdot dS$ if $F= y i+(x-2xz)j- (xy) k$ and S the surface of the sphere $x^2+y^2+z^2=a^2 $ above of the $xy-$plane I do not know if I must use the stokes theorem or try ...
1
vote
3answers
70 views

Find the area between the cylinder $z^2+y^2=r^2$ and two planes

I'm having trouble with this problem: Find the surface area between the top of $z^2+y^2=r^2$ between $z=ax$ and $z=bx$ (consider $a \gt b \gt 0$). I think I must find the area between the ...
0
votes
0answers
6 views

Integration of a dyadic vector over volume

I study a paper which has three lines that I can't reproduce. So, we have integral $\Bigg[\int \frac{(\partial f_0(v)/\partial v) \vec{v}\vec{v}}{v(\omega - \vec{k}\cdot\vec{v})} d\vec{v} \Bigg] \...
0
votes
1answer
45 views

Velocity profile in a triangular duct (Derivation)

How do I find a velocity profile of an incompressible fluid in a triangular duct. Can someone point me to a step-by-step solution so that I could understand the process of derivation of the final form ...
0
votes
0answers
26 views

C alculating flux using the divergence theorem when the divergence is 0

I calculated the divergence of my vector field $\langle x^2 + y^2, y^2 + z^2, 1 − 2xz − 2yz\rangle$ to be $0$. The flux is meant to be over the unit hemisphere. If I do use the divergence theorem, ...
1
vote
1answer
26 views

Calculating the moment of inertia with respect to z-axis (volume integral)

Question states: Consider a body with a surface defined by $2(x^{2}+y^{2})+4z^{2}=1$. Calculate the moment of inertia with respect to the z-axis, i.e. $I_{z}=\int\int\int_{V}(x^{2}+y^{2})dxdydz$. I ...
0
votes
0answers
49 views

Partial derivative of vector intercepting a plane

I was reading a paper that describes the partial derivatives of a range $\rho$ that intercepts an arbitrary surface, where $\rho = |\bar{r}_t - \bar{r}_{bf}|$. The author described the influence of an ...
0
votes
0answers
18 views

Calculating x and y coordinates from curvilinear orthogonal coordinates.

We have a curvilinear orthogonal coordinate system defined with $u=xy$, $v=\frac{x^2 - y^2}{2}$, $z=z$. First, calculate x and y. For them, I got $x=\sqrt{\frac{-2v\pm2\sqrt{v^2-u^2}}{2}}$ and $y=u/x$...
1
vote
1answer
48 views

Calculating the divergence of the Gravitational field $\nabla \cdot \vec{F}$

I want to calculate the divergence of the Gravitational field: $$\nabla\cdot \vec{F}=\nabla\cdot\left( -\frac{GMm}{\lvert \vec{r} \rvert^2} \hat{e}_r\right )$$ in spherical coordinates. I know that ...
0
votes
0answers
29 views

Calculate the flux of $F(x,y,z) = 3xy^2i + 3x^2yj +z^3$ on unit sphere?

I can't seem to work out this problem: Find the flux of $F(x,y,z) = 3xy^2i + 3x^2yj +z^3k$ out of the unit sphere centred at (0,0,0). My attempt is as follows: \begin{align*} \iint_S F \cdot dS &...
1
vote
2answers
33 views

How do I calculate gradient?

$q(x)=x^TAx+b^Tx+c$ Where A is matrix, $x,b\in \mathbb{R}^n $ $c\in \mathbb{R}$ So someone in my book wrote that q(x) is the same like $q(x)=a_{11}x_1^2+...a_{nn}x_n^2+2a_{12}x_1x_2...+2a_{ij}...