Questions tagged [vector-analysis]

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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Chain rule for multivariable differentiation

For one-dimensional case, after affine transformation \begin{equation} x=\hat{x}+\sigma c . \end{equation} Using the chain rule for differentiation, we can write \begin{equation} \frac{d}{dc}=\frac{...
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35 views

Suitable books for further study on mathematical analysis

I have self-studied mathematical analysis using books written by people from my country. These books cover traditional topics including green formula and stokes formula ($\mathbb{R}^3$ form). However,...
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Flux through the curved surface of the cylinder in the first octant.

I am given a vector field $\vec{F}=z\hat{i}+x\hat{j}+y\hat{k}$. The flux of the vector field passing through the curved surface of the cylinder $x^2+y^2=a^2$ in the first octant and below $Z=h$ plane ...
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1answer
41 views

Question about expression in Griffiths

So, in Griffiths' E&M book, he comes up with this expression for the magnetic dipole moment, $$A_{\text{dip}}(\textbf{r}) = \frac{\mu_{0}I}{4\pi r^{2}} \oint r'\cos(\alpha) \,d\textbf{l}' = \...
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Spherical Integral from a dot product in d dimensions

I was doing some integrals in arbitrary dimensions, and I know the answer I am supposed to get, but I cant seem to get there. The integral is $$\int d^D k e^{i\vec{k} \cdot \vec{r}} \frac{1}{k^2+m^...
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73 views

Assistance with a Vector Calculus Exercise

I am computing $\nabla \nabla$ of a function $\mu_{\gamma}=-\frac{e^{-\gamma}}{4 \pi r}$, and get the following (since the function only depends on $r$). $$ \begin{split} \nabla \mu_{\gamma} &= \...
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Proof of divergence theorem

I am trying to proof the classical Gauss's divergence theorem given by, For a bounded domain $D\in \mathbb{R}^N$ with a smooth boundary $\partial D$. For the function $F=(F_1,F_2,...,F_N)$ it is ...
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Find the gradient of $cf(x^Tg(z))$ with respect to x

As I understand it should be something like this: $$\dfrac{\partial y}{\partial x}=(cf'g(z))^T$$ but I saw that $$(g(z)cf')^T$$ Which one is correct? maybe none..
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63 views

Surface Area of $n$-Dimensional Sphere with Multiple Hyperplanar Cuts

Let $S^{n-1}\subset\mathbb{R}^n$ be the unit sphere, and $v_1, \cdots, v_m\in S^{n-1}$ be $n$-dimensional unit vectors. Each of these vectors defines an $n-1$ dimensional hyperplane, which cuts $S^{n-...
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Why does Gauss divergence give different answer?

If $ \vec{F} = y\hat{\imath} + 2x\hat{\jmath}-z\hat{k}$ and S is a surface of plane $2x+y=4$ in the first octant cutoff by plane $z=4$ then evaluate $\int_S \vec{F}\hat{n}dS$ Now if I solve this ...
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I am getting error while verifying Gauss Divergence Theorem.

When I am verifying divergence theorem for $\vec{F}=(x+y^2)i-(2x)j+(2yz)k$ , surrounded by the co-ordinate planes and the plane $2x+y+2z=6$ , I am getting some errors. While calculating the using the ...
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Tips on studying and becoming proficient at vector calculus

I am a computer science undergrad. I am pursuing my second year open university BS mathematics course. I am currently taking a course on Vector calculus. I am using two text-books - Vector analysis by ...
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28 views

How do I check that my edge normals in an arbitary triangle are inward (or outward)?

I am considering an arbitary 2D triangle with points: X1=(x1,y1), X2=(x2,y2), X3=(x3,y3). I am then trying to compute the inward (or outward) normals to each edge such that they are all facing ...
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44 views

Exercise showing that $\nabla f(r) = \frac{\mathrm{d} f}{\mathrm{d}r}\cdot \frac{\underline{r}}{r} $

Let $f : \mathbb{R} \rightarrow \mathbb{R}$ be a differentiable function. Let $$\underline{r}(x,y,z) := \begin{pmatrix}x\\y\\z\end{pmatrix}$$ be a vector field in Cartesian coordinates. The length $...
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Grad in cylindrical coordinates.

I'm trying to work out the gradient of a function in cylindrical coordinates. With $x=\rho\cos\phi$, $y=\rho\sin\phi$ and $z=z$, the Euclidean metric is $$g=d\rho^2+\rho^2d\phi^2+dz^2$$ And the ...
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Is the gradient vector tangent to the surface?

I understand the reason why the gradient vector is always orthogonal to the level sets of f, but I just cannot find any notes saying the gradient vector is tangent to the surface. But it seems to be ...
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82 views

Prove $\oint_\Gamma\vec\nabla f\cdot d\vec{r}=0$ when $\Gamma$ is the unit circle

I have encountered a statement in my book which didn't seem quite right to me. It was written exactly like this: Let $f(x,y):\mathbb{R}^2\to\mathbb{R}$ be a differentiable function such that $\frac{...
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1answer
36 views

Gradient formula in Lee Smooth Manifolds differs from others?

In example 13.31 (page 343) of Introduction to Smooth Manifolds, Lee uses the musical isomorphisms to calculate the gradient in polar coordinates. He obtains: $$\text{grad} f = \frac{\partial f}{\...
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1answer
18 views

Parametric equations of straight line in cylindrical and spherical coordinates

The most general equation of a straight line is $$\vec{r}=\vec{r}_0+\vec{a} s$$ where $s$ is a parameter, the line passes through point $\vec{r}_0=(x_0,y_0,z_0)$ and is parallel to vector $\vec{a}=(...
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27 views

Why is $d\mathbf{a}$ a vector when doing surface integrals?

I'm just learning about surface integrals, and I'm doing a problem where I'm supposed to calculate the surface integral of some vector function $\mathbf{v}$ over five sides of a cube. For the first ...
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1answer
26 views

Confused with the surface integral in the proof of the divergence theorem

I know how to work with the triple integral of the divergence of F part of the theorem, but in many textbooks, they don't explain the surface integral component. I don't understand how they go from ...
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1answer
28 views

how to convert surface to vector field to use it in line integral

example for what i want to say how to convert the scalar form of line integral :- $\int g(x,y)\ ds = \int p\ dx + \int q\ dy$ real world example:- $\int xy \ ds \ s: unit \ circle \ centered at ...
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How to compute $(\mathbf{A} \cdot \mathbf{\nabla})\mathbf{B}$?

I'm currently reading Intro to Electrodynamics by Griffiths, and in the maths section, there is the following problem: "If $\mathbf{A}$ and $\mathbf{B}$ are two vector functions, what does the ...
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48 views

How to prove that the stable equilibrium has a jacobian matrix of which all eigenvalues are negative OR zero?

Sorry to be sloppy the first time. Now I have corrected something according to the comments. Let's consider this system of $n$ linear second order ODEs: $$\ddot {\mathbf{x}}=f(\mathbf{x}),$$ Where $f:...
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24 views

Divergence of a tensor product

Let ${A}\in\mathbb{R}^{3\times3}$ be a second-order tensor, for which $\mathrm{tr}{A}=0$, ${{A}}^\mathrm{T}={A}$, and $\alpha\in\mathbb{R}_+$. Is the derivative $\partial_{\nabla A}(\nabla\alpha\...
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64 views

Compatibility condition for Poisson's equation - Divergence theorem

Let $\Omega \subset \mathbb R^n$ be open and bounded. Given the Poissons equation with Neumann-boundary, e.g. $$-\Delta u = f \quad \text{and}\quad \frac{\partial u}{\partial n}=g, $$ we must require ...
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10 views

Pseudo-gradient vector field

According to this question Understanding the idea of a pseudo-gradient vector field, I have another question. In a book I found the following "explanation": [...] as a substitute for the notion of ...
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Understanding a vector analysis formula for finite volume method

I have an equation which is used for the finite volume method in partial differential equations. I can read it but I had to few vector analysis to really get it. The equation is: \begin{equation} \...
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Proving identity related to tangent and binormals

If unit tangent vector $t $, and binormal vector $b$, makes angle $\theta$ and $\phi$ respectively with constant unit vector $a$, prove that $$\frac{\sin\theta\,d\theta}{\sin\phi\, d\phi} = \frac{-k}{\...
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1answer
43 views

Finding length of cross product of two vectors

Given $ v = (-2a\sin(2t),2a\cos(2t), 2a\cos(t) )$ and $k= (-4a\cos(2t),-4a\sin(2t),-2a\sin t)$ Finding $|v \times k|$. Now I am finding it hard to compute determinant. There must be an easy way ...
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Finding limits of integration for curve

Given curve is $x^2+y^2=1,~z=1$ from $(0,1,1)$ to $ (1,0,1)$. I am required to find line integral over some $F(x,y,z)$. I have parametrized the curve as $r= (\cos(t),\sin(t),1)$ and value of $t$ goes ...
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Application of the products of four vectors

I am taking a course on Vector calculus in my second-year open university BS Maths course. For rigorous self-study, I am using the book Vector Analysis by Louis Brand while also referring to other ...
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2answers
52 views

Prove that $\triangledown^2 f(r) = \frac{d^2f}{dr^2}+ \frac{2}{r} \frac{df}{dr} $

$\triangledown^2 f(r) = \frac{d^2f}{dr^2}+ \frac{2}{r} \frac{df}{dr} $ where $r^2 = x^2 + y^2 +z^2$ Now i write as $\triangledown f(r) = f'(r) \frac{r}{|r|}$ . taking again $\triangledown f'(r) \...
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49 views

Prove $\operatorname{ div} (\phi A) = (\operatorname{ grad} \phi) \cdot A + \phi \operatorname{ div} A$

Prove $$\operatorname{ div} (\phi A) = (\operatorname{grad} \phi) \cdot A + \phi \operatorname{ div} A$$ where $A$ is differentiable vector function and $\phi $ is differentiable scalar function. ...
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1answer
24 views

Calculate the flux of a field through a surface

Given the surface $S\{(z,y,z)\in \mathbb{R} | x^2+y^2=1, 0\leq z\leq 1\}$ and the vectorial fields $X(x,y,z)=\frac{1}{x^2+y^2}(x,y,0)$ and $Y(x,y,z)=\frac{1}{(x-3)^2+y^2}(x-3,y,0)$ I have to calculate ...
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How can Green's Theorem be used to derive Maxwell's equations?

I've learned how to prove Green's Theorem and I read that it contributed to deriving Maxwell's equations. How can Green's Theorem be used to derive any of four Maxwell's equations? What else do I have ...
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1answer
34 views

Gauss/Divergence Theorem special case

In one of my calculus exercises, I am asked the following question: Use the Gauss theorem to show that if B is a vector field defined on a open set U in $R^{3}$ that has $div(B)=0$ then the flux of ...
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1answer
32 views

Finding A point $Q$ on a surface such that Line $PQ$ get tangent to that surface

Consider the surface $f: x^2 + y^2 - 2z^2 = 1$ and Point $P= (1,1,1)$. We want to find all points $Q$ on the surface such that line $PQ$ get tangent to the surface. Also we want to find the point $...
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2answers
60 views

Evaluate using Stokes' Theorem

To evaluate $\oint_{C} -y^3dx+x^3dy+z^3dz,$ where $C$ is the intersection of cylinder $x^2 + y^2 =1$ and plane $x+y+z=1$. The orientation of $C$ is counter-clockwise motion in the $xy$ plane. Now I ...
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2answers
46 views

Find a Curve that is perpendicular to another curve

We want to find a curve in $xy$ plane that passes through $(1,1)$ is perpendicular to all of contour curves of form $f(x,y) = x^4 + y^4$. For the solution, I considered the curve as $r = (x(t) , y(t))...
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1answer
23 views

Finding tangent to intersection of two surfaces

In $\mathbb{R}^3$ we have two surfaces as follows: $$f(x,y,z)= x^2 + y^2 -2 = 0 ~~~(cone)$$ $$g(x,y,z) = x+z-4 = 0 ~~~ (plane)$$ We want to find the tangent line to the intersection of these two ...
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1answer
78 views

Write the functional derivative of the Dirichlet energy

I have the following functional: $J(u) = \int_{\Omega} \frac{|\nabla u|^p}{p}\,d\Omega$ and I want to compute its functional derivative along the direction of an arbitrary test function $v\in H_0^1$....
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1answer
23 views

For a solenoid vector field $F$, show that $\nabla\times(\nabla\times(\nabla\times(\nabla\times(F))))=\nabla^{4}F$

For a solenoid vector field $F$, show that $\nabla\times(\nabla\times(\nabla\times(\nabla\times(F))))=\nabla^{4}F$ Now we have $\nabla \cdot \vec F=0$ : Therefore $\nabla\times(\nabla\times \vec ...
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1answer
47 views

Intuition on gradient in other coordinate system [duplicate]

I know that gradient formula in other coordinate system is more complicated than the Cartesian formula. For example, the gradient in polar coordinate system is $$ \nabla f = \frac{\partial f}{\...
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1answer
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Why is Kullback–Leibler divergence a divergence. How is it related to divergence of a vector field? [closed]

Why is Kullback–Leibler divergence a divergence. How is it related to divergence of a vector field
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Gradient approximation on non-uniform quadrilateral mesh

I have non-uniform quadrilateral mesh. For each volume from my mesh I have a point which represents the volume. I need gradient approximation using representation points. Any ideas?
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2answers
27 views

Difference between $\nabla(\nabla . F)$ and $\nabla^2F$, where $F$ is a vector function

I recently started vector calculus and saw that following identity $\nabla\times(\nabla\times F)=\nabla(\nabla.F)-\nabla^2F$ But aren't $\nabla(\nabla.F)$ and $\nabla^2F$ the same or I am missing ...
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37 views

Least square method, why gradient gives the minimum and not the maximum?

I'm bit confused about finding critical points of functions. Studying the least square method we got some data $\{(y_{1},x_{1}),...,(y_{n},x_{n}))\}$ and can define error associating with $y=ax+b$ by ...
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82 views

The geometric interpretation of the gradient for a function written in quadratic form

Alright, this question is going to be really broad, and that's my fault because I am extremely lost. I appreciate all the help! As far as I understand, writing a quadratic multivariable function in ...
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0answers
20 views

Gradient of squared euclidean norm

For a given: $A \in \mathbb{R}^{m \times n}$ I am trying to calculate the gradient of: $$f(x) := \frac{1}{2}\left\|{A^T x}\right\|_2^2 - b^Tx.$$ Therefore I wrote $f$ like this: $$f(x) = \frac{1}{...