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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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Gradient of $f(x,y)=\sqrt{x^2(1+y)+y^2+4}$

I have to solve the gradient of $f(x,y)=\sqrt{x^2(1+y)+y^2+4}$ as a part of a larger task. I know how to do this with partial derivatives but I was wondering if there are simpler ways to find the ...
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1answer
17 views

Calculating the dot product of vector fields.

Given Two vector fields : $$X(x,y,z)=(x,-y,-z)$$ $$Y(x,y,z)=(1,-y,x)$$ I want to calculate the dot product of these two vector fields $X.Y$. It's just that its vector fields that is confusing me a ...
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2answers
32 views

Find partial derivatives of $f(x)=\|x\|^\alpha$

Find partial derivatives of $f:\mathbb{R}^n\rightarrow\mathbb{R}$ $$f(x)=\|x\|^\alpha$$ outside of $(0,0)$ when $\alpha\in\mathbb{R}$. What values does $\alpha$ have to take for the partial ...
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vote
1answer
38 views

A chain rule for the angle

My question is fairly basic, namely can I do the substitution below? $$ \frac{\mathrm{d}\theta(t)}{\mathrm{d}t} = -\frac{1}{\sin(\theta(t))}\frac{\mathrm{d}\cos(\theta(t))}{\mathrm{d}t} $$ If not, ...
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0answers
11 views

How to write basis vectors in Cartesian coordinates in terms of cylindrical coordinates?

Given a Cartesian coordinate system with basis vectors (ex, ey, ez) and a Cylindrical coordinate system with basis vectors (er, eθ, ez) Why and how does: ex = erCos(θ) - eθSin(θ) ? ey = erSin(θ) +...
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2answers
51 views

Prove that the $2Area(\omega) \leq length(\gamma)distance(\gamma)$

So basically, I am given the following to prove: Let $+\gamma$ be a positively oriented smooth Jordan arc, and let $\omega$ denote the interior of $+\gamma$. Recall that if $F = (F_1, F_2):D \to \...
2
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1answer
30 views

DGL: variational problem in $H^1$

I've given $f \in L^2(\Omega), g \in H^1(\Omega)$. I want to find $u \in H^1(\Omega)$ such that $$ -div (A \nabla u ) + <b, \nabla u> + cu = f$$ in $\Omega$ $$u = g$$ on $\Gamma$. In order to ...
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2answers
41 views

How to evaluate $\nabla \frac 1r$, $\nabla^2 \frac 1r$, $\int_S \nabla \frac 1r . ndS$

Let S be a smooth closed surface in a three-dimensional xyz-space, n, be the unit outward normal vector on S, and r be the distance between the origin and a point (x, y, z). Solve the following ...
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1answer
12 views

Deriving the area of surface of revolution formula from cylindrical coordinates

I'm given a surface $S = \{(r, \theta, z): a \leq z \leq b, r = g(z)\}$ in cylindrical coordinates. Now I have to derive the area of surface of rotation which is $\int_{a}^{b}2\pi g(z)\, \sqrt{\,1 + (...
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1answer
58 views

Show $\frac{df}{dz}=\frac{\hat{r}\bullet\nabla f}{e^{j\phi}}$ . Where $z$ is a complex number and $f$ is differentiable at z.

Show $\frac{df}{dz}=\frac{\hat{r}\bullet\nabla f}{e^{j\phi}}$ . Where $z$ is a complex number and f is differentiable at $z$. The $\bullet$ denotes the dot(inner) product. $\nabla$ is the gradient. $...
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0answers
54 views

if $\int_{\mathbb R} f(x) dx \cdot v $ converges, is it possible that $\int_{\mathbb R} f(x)\cdot v\ dx $ diverges?

Placing a dot product inside the vector valued integral can "tame" bad integrands. A toy example is if $\operatorname{bad}(x)$ is some bad integrand, maybe like $-x^{100} + \frac{1}{x^{100}}$, then ...
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1answer
28 views

A inequality between $||u||_{p}$ and $||\gamma (u)||_{p, \partial \Omega}$, where $\gamma$ is the Trace Operator?

Does someone know any inequality between $||u||_{p}$ and $||\gamma (u)||_{p, \partial \Omega}$, where $\gamma$ is the Trace Operator? I need to find something like $||u||_{p}\leq C||\gamma (u)||_{p, \...
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1answer
21 views

Why is the gradient defined as a vector function and not as a vector?

According to this article: The gradient of a function w=f(x,y,z) is the vector function: $<\dfrac{\partial f}{\partial x} (x, y, z), \dfrac{\partial f}{\partial x} (x, y, y),\dfrac{\partial f}{\...
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0answers
19 views

Iterative Vector Matrix Product

Consider the following iterative matrix product: $x_{i+1} = A_{i} \cdot x_{i}$. The matrices $A_i$ are defined as follows: All rows except of the first and last row are stochastic. The first and the ...
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1answer
23 views

Calculate variations of eigenvectors w.r.t input matrix

From the eigen decomposition : $$A = PDP^T$$ I would like to calculate $dP$ w.r.t $dA$. I start like this : $$dA = dPDP^T + PdDP^T + PDdP^T$$ Since we only consider variations in eigenvectors and ...
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0answers
12 views

Conditions to tell that the total flux in a vector field across a closed surface is zero

When is the total flux across a closed surface zero. I am trying to find a set of values for an equation to prove that it has a total flux of 0 across a closed surface. I am not given any specific ...
3
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1answer
73 views
+50

Points of Confusion About Second-Order Taylor Formula of Taylor's Theorem For Many Variables

My textbook has written the following for the second-order Taylor formula of Taylor's theorem for many variables: $$f(\mathbf{x}_0 + \mathbf{h}) = f(\mathbf{x_0}) + \sum_{i = 1}^n h_i \dfrac{\...
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0answers
23 views

What is the gradient of a phase space density function?

For a phase space density function $f(\vec{r}, \vec{v}, t)$, where $\vec{r}$ and $\vec{v}$ are $n$-dimensional vectors (in this case position and velocity), is the gradient of $f$ a $2n$-dimensional ...
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1answer
40 views

Why is the flux of a conservative field always positive?

I have been given this question as a task. I have done some mathematical analysis but they were of no help. The only thing I could think of was that total flux in a closed surface can be calculated by ...
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2answers
86 views

Differentials in Multivariable Calculus

Does the idea of composing/decomposing the fraction notation of the derivative from/into differentials apply in multivariable calculus? I realize that this practice is considered non-standard and ...
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1answer
40 views

Calculating Divergence $\nabla_\mu V^\mu$ and Laplacian $\nabla_\mu\nabla^\mu f$

If I have just these three equations \begin{align*} x = uv\cos\phi,\quad y = uv\sin\phi,\quad z = \frac{1}{2}(u^2-v^2) \end{align*} I'm asked to find the divergence $\nabla_\mu V^\mu$ and ...
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1answer
88 views

Showing that the unsigned area bounded by plane curve $\gamma$ is $-\frac1{4\pi}\oint_\gamma\oint_\gamma\vec{dx}\cdot\vec{dy}\log(\|x-y\|^2)$

Let $\gamma$ be a curve in the plane. I wish to show: $$A=\frac{-1}{4\pi}\oint_\gamma\oint_\gamma\overrightarrow{dx}\cdot\overrightarrow{dy}\log\left(\|x-y\|^2\right),$$ where here $A$ is the ...
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0answers
13 views

Invariance of nabla under pure rotation

The problem goes as follows: Show that, under a rotation $\vec {\nabla} = \hat{i} \frac{\partial}{\partial x}+ \hat{j} \frac{\partial}{\partial y}+ \hat{k} \frac{\partial}{\partial z} = \hat{i'} \frac{...
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0answers
23 views

Validity of axes renaming

The position vector is $\vec{r}=(x,y,z)$. $\vec{A}$ is a vector field, $\vec{A}=(a_1,a_2,a_3)$ and $C$ is a constant. We have the function $$f=f(a_1x,a_2x,a_3x,Cx^2).$$ Can one then say, that ...
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0answers
27 views

Intuition on mean curvature being equal to divergence of normal vector

For starters, such a thing as the divergence of the normal vector is already "counter intuitive" to me because of the definition of divergence I like using (local flux density, involving an ...
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0answers
23 views

Derivative of a scalar given by a function times its hessian

Is there any way of doing the below but by avoiding $d^3f\over dx^3$? If for a row vector $x$ e.g. size (3x1): $f=f(x)=scalar$ $g={df\over dx}$ i.e. the gradient $H={d^2f\over dx^2}$ i.e. the ...
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1answer
69 views

Show a vector identity including derivative $a\cdot \frac{\partial F}{\partial q}b=\left(\frac{\partial F}{\partial q}\right)^T a\cdot b$

Can somebody help me to show: $$\boldsymbol{a\cdot}\frac{\partial \boldsymbol{F}}{\partial \boldsymbol{q}}\boldsymbol{b}=\left(\frac{\partial \boldsymbol{F}}{\partial \boldsymbol{q}}\right)^T \...
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2answers
27 views

How to find the angle bisectors of 2 lines

I tried gooogling how to do it but there were none that could relate to the problem I have. Find the angle bisectors of the lines: $$g: r= \begin{pmatrix} 2 \\ 5 \\ -9 \\ \end{...
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1answer
38 views

Covariant and contravariant of curvilinear system with given vectors

I am given a curvilinear coordinates $a$, $b$ and $\phi$ as follows: $$a=\frac{x^2+y^2+z^2}{2z},\ b=\frac{x^2+y^2+z^2}{2\sqrt{x^2+y^2}},\ \phi=\arctan\frac{y}{x}.$$ I am asked to find covariant and ...
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1answer
22 views

Relation between transformation matrices and conversion formulas between coordinate systems?

We are learning how to work with different coordinate systems in my Mechanics class (spherical and cylindrical mainly), and about form factors, general formulas for the gradient, the curl, the ...
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2answers
51 views

Derivative of a vector times its transpose

I am trying to work out how to solve a derivative of the form: $${d \over dx}(M(x)M(x)^T)$$ where M is a vector. In my case specifically, M is the (1x3) vector $$M(x)={df(x) \over dx}$$ where f(...
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3answers
58 views

Showing that $∇·(s∇×(s\vec V))=(∇×\vec V)·∇(s^2/2)$

I was given to prove $$∇·(s∇×(s\vec V))=(∇×\vec V)·∇(s^2/2)$$ by using index notations i.e. Kronecker delta and Levi-Civita symbol. But I cannot figure out how I have to tackle the problem to get that ...
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2answers
35 views

Equation of tangent plane to a parametrised surface

I've got a problem trying to figure out what I'm doing wrong with these question regarding finding the equation of the tangent plane to a parametrised surface. A surface is parametrised by $$x = u^2-...
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1answer
28 views

When a third vector in a plane does not lie in the span of 2 linearly independent vectors in the plane

For instance, can the 3 vectors $\vec a=[1, \ 0, \ 1]^T, \vec b=[2, \ 7, \ -2]^T, \vec c=[3, \ 1,\ 5]^T$ lie on the same plane in $\mathbb R^3$? My understanding is that the span of 2 linearly ...
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1answer
19 views

Curve $\alpha(t)$ with evolute of $E(t)=(t,1/t)$

I've been trying to independently learn differential geometry/vector calculus, and I've been experimenting with evolutes and curvature. I came up with the following problem that I need help solving. ...
2
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2answers
46 views

Motion of an object that always has an acceleration perpendicular to its velocity

Consider an object whose position vector $x$(changes with time) satisfies the condition $$ \dot{x}\cdot\ddot{x}=0 $$ i.e. the object is always accelerating in the direction perpendicular to its ...
2
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1answer
44 views

How to prove the distributive law of dot product (i.e $A.(B+C) = (A.B) + (A.C)$)when three vectors $A , B , C$ are not in the same plane?

How to prove the distributive law ( i.e $A.(B+C) = (A.B) + (A.C)$) when three vectors $A , B , C$ are not in the same plane? I was reading the chapter called Multiplication of vectors. I had a hard ...
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0answers
36 views

Find the partial derivatives of the function $f(x)=\|x\|^{\alpha}$ outside of $(0,0)$

We had the following solved problem in our lecture which I didn't quite fully understand. Find the partial derivatives of the function $f:\mathbb R^n\rightarrow \mathbb R$ $$f(x)=||x||^{\alpha}$$ ...
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2answers
19 views

Is $P$ a subspace of $\mathbb{R_{≤2}[x]}$?

I have a polynomial function defined as :$$P:= (p \in \mathbb{R_{≤2}[x] \ | \ p(0)=0})$$ and I want to prove that $P$ is a subset of $\mathbb{R_{≤2}[x]}$. I use the $3$ criteria that need to be ...
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1answer
26 views

Find the second rank tensor that satisfies the identity.

I was given the following problem to consider: Find the rank II tensor T that satisfies the identity, $T - Tr(T)\delta = \delta\times\vec{a}$ No information was given for the vector $a$, so we ...
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0answers
38 views

Injective mapping theorem

I have been searching for a proper explanation to " The Injective Mapping Theorem " in R. G. Bartle which deals with derivatives of vector functions, but most of the results I found till now gives ...
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0answers
58 views

Please help, I just have a simple question about curves

I've been experimenting with the curvature of parametric curves on Desmos, and have gotten the following results. I cant really explain my question with words, so here's a bit of an introduction to my ...
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1answer
32 views

Expressing Vectors, Finding the Angle and Finding the Area

Part 1 According to the Rectangular solid in the following image, express the vectors $\overrightarrow {AF}$, $\overrightarrow {GD}$, $\overrightarrow {FC}$, $\overrightarrow {EC}$ in terms ...
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1answer
32 views

Multiple minima points and no maximum

Question: Use the bordered Hessian test to show that $f(x,y,z)=x^2+y^2+z^2$ under $g(x,y,z)=z-xy-2=0$ has two minimum points and no maximum (find the two points), and explain how this is possible. I'...
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0answers
62 views

Examples of Tensor Transformation Law

Let $T_{\mu\nu}$ be a rank $(0,2)$ tensor, $V^\mu$ a vector, and $U_\mu$ a covector. Using the definition of tensors based on the tensor transformation law, determine whether each of the following is ...
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1answer
64 views

Does $ \vec x=\begin{pmatrix}1 \\ 0 \\ 1 \end{pmatrix}$ lie in the Image of L

I have the following linear transformation: $$L: \mathbb{R^3} \to \mathbb{R^3}, \begin{pmatrix}a \\ b \\ c \end{pmatrix} \mapsto \begin{pmatrix}a-c \\ 0 \\ b \end{pmatrix}$$ and I have to check ...
1
vote
1answer
32 views

Surface integral through a cube.

I am doing some fluid dynamics work and the question asks: Consider the mass flow vector: $\rho \vec{u} = (4x^2y,xyz,yz^2)$ Compute the net mass outflow through the cube formed by the planes x=0, ...
2
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0answers
76 views

Vector field in $\Bbb Q^2$ and $\Bbb R^2$

I'd like to define a weak vector field in $\Bbb R^2$ that is tangential to a family of sine curves at each point. I define the family of sine curves as: $A_n\sin(x)$; $n=1,2,3,...$ and $A_n$ is a ...
0
votes
0answers
49 views

$\frac{d}{d x_1} \int_{-c}^{F(x_1,x_2)}v_1(x_1,x_2,x_3)dx_3\overset{!}{=}-v_1(x_1,x_2,F(x_1,x_2))\frac{\partial}{\partial x_1}F(x_1,x_2)$

This equation is part of a proof of the divergence theorem in $\mathbb{R}^3$ for sets $$M=\bigg\{x \in \mathbb{R}^3: x_3 \leq F(x_1,x_2) \bigg\} \cap[0,1]^2\times[-c,c]$$ in which $F: [0,1]^2 \...
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0answers
22 views

Question Regarding Vector Calculus

Is $(\vec{\nabla}\cdot\vec{f})\vec{f}=\vec{f}(\vec{\nabla}\cdot\vec{f})$? I know that the $i$-th components of the left hand side and right hand side are $(\partial_jf_j)f_i$ and $f_i(\partial_jf_j)$ ...