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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14 views

Proof of a sustitution on the magnitude of the separation vector.

Given the separation vector and its magnitude: $\displaystyle \vec{r} =( x-x') \ \hat{i} +( y-y') \ \hat{j} +( z-z') \ \hat{k}$ $\displaystyle r=\left[( x-x')^{2} +( y-y')^{2} \ +( z-z')^{2}\right]^{...
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21 views

Points A, B and C have position vectors a = i + 2 j, b = 2i + j and c = 2i − 3 j. Find the shortest distance from B to line AC

Points A, B and C have position vectors a = i + 2 j, b = 2i + j and c = 2i − 3 j. Find: The shortest distance from B to line AC The area of triangle ABC What I have done so far: Found AB and ...
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26 views

How to Prove a Special Case of Stokes' Theorem?

I am currently in Calculus 3, or Multivariable Calculus and need to prove this special case of Stokes' theorem. Please forgive me as I do need this simplified to the bones to understand the ...
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25 views

Convert first order ODE system into a complex ODE

Consider the first-order ODE system for the real 2D vector $X = (x,y)$ $$ \frac{d}{dt} X(t) = (r + I) \nabla U = r \nabla U + I (\nabla U) $$ where $I$ is the $2\times 2$ rotation matrix of $\pi/2$ ...
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26 views

How is obtained the formula $\int_\Omega \Psi \text{div} F dx =\int_{\partial \Omega }\Psi F \cdot \nu dS - \int_\Omega \nabla \Psi \cdot Fdx $

How to show that this inequality holds? $\int_\Omega \Psi \text{div} F dx =\int_{\partial \Omega }\Psi F \cdot \nu dS - \int_\Omega \nabla \Psi \cdot Fdx $ Where $\psi$ is a scalar function and $F$ ...
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30 views

Why this does not have a dot product on the right: $\int_\Omega \nabla \phi dx =\int_{\partial \Omega}\phi \nu ds$

My main concern is because, on the right side of the expression, there is a vector $\nu$, but there is not any dot product. $$\int_\Omega \nabla \phi dx =\int_{\partial \Omega}\phi \nu ds$$ Where $\...
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22 views

Flux through cone

Consider the vector field $\mathbf{v}=(x^2y-\sqrt[3]{z}, \sqrt[3]{z}-xy^2,z)$ and a surface given by $\frac{z^2}{2}=x^2+y^2$ and $z\in[0,2]$. Calculate the flux of $\mathbf{v}$ through the surface. ...
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27 views

Unit vector for the minimum directional derivative of a function

For the following function: $f(x,y)=\ln(x-y)$, at $(2,1)$ what is the unit vector for the minimum directional derivative of $f$ ? Firstly to find the gradient of $f$ at $(2,1)$: $$ \nabla f(x,y) = \...
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35 views

The above vectors is taken to be with respect to the standard basis for $\mathbb{R}^2$?

I am currently studying Introduction to Tensor Analysis and the Calculus of Moving Surfaces by Pavel Grinfeld. Exercise 3 of Chapter 1 Why Tensor Calculus? proceeds as follows: Exercise 3. Show ...
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Let $S$ be the graph of a function of class $C^{1}$ over a set $A\in\mathbb{R}^{2}$ Then its area is given by $\int_{A}sec(\alpha)dxdy$

Let $S$ be the graph of a function of class $C^{1}$, $z=g(x,y)$ over a set $A\in\mathbb{R}^{2}$, then the area of the surface $S$ is given by $\int_{A}sec(\alpha)dxdy$ where $\alpha$ is the angle ...
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21 views

Nonlinear vector calculus problem

Let $A$ be a vector field on $\mathbb{R}^3$. I am interested in finding solutions of $$ \nabla^2 A \times {\rm curl} A = 0,\\ \quad {\rm div} A = 0. $$ Are there any exact solutions with nonzero $\...
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25 views

Cylindrical coordinates, time derivatives, and the Navier-Stokes equations.

I am trying to derive the Navier-Stokes equations in cylindrical coordinates, but am having trouble with the material derivative. Some background. The unit vectors in Cartesian coordinates are given ...
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14 views

Gradient of a cross product of a vector and second rank tensor

Consider a vector $\vec{x}$ in three dimensions and $3\times 3$ second rank symmetric tensor $\textbf{T}$. What is the definition of $\nabla \cdot(\vec{x}\times\textbf{T})$? Based on this answer, I ...
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26 views

Is there a way to add two complex vectors and get both magnitude and phase?

Let us consider a spherical coordinate system and an electric field that has two complex components, $E_\theta$ and $E_\phi$. There is no dependence with $r$. Therefore, at any pair of coordinates $(\...
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How to show If f R^n->R, f is a C1- Class function then f is differentible [closed]

C1 = every partitial difference of f is continuous How do I use f is C1-class condition ?
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Can I find $b$ from $a = b \times c$?

So I just started studying the TNB (or Frenet-Serret) frame, where B = T × N. Then my book also goes on to say that T = N × B and N = B × T. Basically, we can find a new valid cross-product equation ...
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15 views

$\nabla(\vec{A}\cdot\vec{r})=\vec{A}.$ Using index notation for vector calculus. Practicing for exam

I am learning the index notations in vector calculus and tried to prove the following identity: Let $\vec{A}=(A_x,A_y,A_z)$ is a constant vector and let $\vec{r}=(x,y,z)$. Prove that $$\nabla(\vec{A}\...
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Is it possible to make an integral by part on measurable sets?

I'm trying to show that a function $u \in W^{1,p}(\Omega)$, where $p \ge 1, n \ge 1$ and $\Omega\subset \mathbb{R}^n$ a bounded open. Where $u$ can be written as $$\sum_{j=1}^m u_j \mathbb{1}_{A_j}$$ ...
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32 views

Line integration in pauls notes calculus III

Why did he use $[0,2]$ as the limits of integration in the 3ed curve after substituting the value of $x = 1$ in $x$, shouldn't the limits be $[0,1]$ as in the $X$ axis, so the last integration ...
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55 views

Integral of a differential form

I'm reading a note about integral of a differential 1-form and would like to know more about its derivation. Is there an intuitive way to understand this formula? Something akin to "sum of weighted ...
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8 views

What does the surface gradient dot product with the normal vector mean?

The surface gradient is defined as $$ \nabla_s = \nabla - \bar n \frac{\partial}{\partial n} $$ if applied on a function $\sigma$ we get $$ \nabla_s \sigma = \nabla \sigma - \bar n \frac{\partial \...
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38 views

Angle of Inclination if Range is at Maximum

If the horizontal range of a projectile launched, without air resistance is given by: $$R = \frac{v_0\cos\theta}{g}\left(v_0\sin\theta +\sqrt{v_{0}^{2}\sin^{2}\theta + 2S_0g}\right)$$ where $S_0$ is ...
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31 views

Multivariable Calculus, more specifically flux integral

So the question gives vector field $F(x,y,z)=(x^2,y^2,z^2)$ and it is flowing out of a sphere of $(x-1)^2+(y+1)^2+(z-2)^2 \leq 4 $. Using Guass's rule, I transformed it into a triple integral while ...
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25 views

How to find the integration limits of a surface integral involving $z=x+y^2$, $z<0$ and $x>-1$

I am trying to evaluate the surface integral of a vector function u (whose definition is irrelevant to this problem) over the surface $$z=x+y^2 $$ with $z<0$ and $x>-1$. Initially I put $$x+y^...
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1answer
23 views

Why does this “gradient field test” not work on the spin field $S / r^2$? (from Strang's Calculus)

In section 15.2, Strang's Calculus explains that for any gradient field $\bf{F} = Mi + Nj$, ${\partial M \over \partial y} = {\partial N \over \partial x}$. (Strang calls this "test D" for identifying ...
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Existence and uniqueness of solenoidal decomposition $f=f^{\text{s}}+\nabla\phi$ for a vector field $f\colon\mathbb{R}^n\rightarrow\mathbb{R}^n$?

I am interested in the existence of a global solution to the Poisson equation $$ \Delta\phi=g \quad \text{in} \ \mathbb{R}^n $$ such that $\phi\rightarrow 0$ when $|x|\rightarrow\infty$. My motivation ...
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What does it mean to take the “ gradient with respect to the position $r_ i$”?

Let’s say we have a number of particles (charged, massive or anything that can create potential energy). The total potential energy of any particle can be given by $$U_i (\vec r_1, \vec r_2, ... \vec ...
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45 views

Derivative of vector functions [closed]

i am struggling the derivation of the function on f). Does anyone know how to solve that?
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38 views

$\vec{F}=\frac{x\hat{i}-y\hat{j}}{b^2x^2-a^2 y^2}$ be a planar vector field, Find $\oint_C\vec{F}.d\vec{r}$

For $a>0, b>0,$ let $\vec{F}=\frac{x\hat{i}-y\hat{j}}{b^2x^2-a^2 y^2}$ be a planar vector field. Let $$C=\{(x,y)\in\mathbb R^2|x^2+y^2=a^2+b^2\}$$ be a circle oriented anti-clockwise. Then $$\...
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31 views

Green's Theorem to calculate a line integral

Use Green's Theorem to evaluate the line integral $\int_C y^3dx + x^3dy$ where $C$ is the ellipse $\frac{x^2}{9} + \frac{y^2}{25} = 1$, so I did this using symmetry. However, I friend switched to ...
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20 views

Surface integral on close surface

Currently, I was reading this paper "Finite element methods for surface PDEs". For the theorem 2.14 in the paper: $ \int_{\Gamma} \nabla_{\Gamma} f \cdot \nabla_{\Gamma} g \mathrm{d} A=-\int_{\Gamma} ...
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28 views

curve of steepest ascent

suppose I have $$z=f(x,y)$$ differentiable in every point. I want to find a parametric curve along which we will get the steepest ascent of the plane. I know that for a given point the direction ...
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Information/references/examples on fields $\mathbb R^3 \to \mathbb C^3$ with divergence and curl free real and imaginary parts

In the course of some physical considerations I came across a complex vector field $$ \mathbf u = \mathbf v + i \mathbf w, $$ with \begin{align} \mathbf v:& \mathbb R^3\to \mathbb R^3\\ \mathbf ...
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63 views

Line integral of the divergence of a curve's unit tangent vector.

I've supplied an answer to another question here https://mathematica.stackexchange.com/questions/222064/line-integration-given-tangent-vector/222205#222205 I'd like somebody more familiar with vector ...
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17 views

$\vec{F}=(3+2xy)\hat{i}+(x^2-3y^2)\hat{j}$ and $L$ be the curve $\vec{r(t)}=e^t\sin t \hat{i}+ e^t \cos t \hat {j}.$ Then $\int_{L}\vec{F}.d\vec{r}$

Let $\vec{F}=(3+2xy)\hat{i}+(x^2-3y^2)\hat{j}$ and $L$ be the curve $\vec{r(t)}=e^t\sin t \hat{i}+ e^t \cos t \hat {j}.$ Then $\int_{L}\vec{F}.d\vec{r}$ My attempt How do I do the other integrals?
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$\vec{y}=\vec{a}\times \vec{x}$. is it possible to define $\vec{x}$ as function of $\vec{y}$

$let\quad\vec{a} \in \mathbb{R^3}$ is it possible to define $\vec{x}$ as function of $\vec{y}$? $$\vec{y}=\vec{a}\times \vec{x}$$ So according to the solution the answer is not and I would like to ...
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difference in computation of gradient and jacobian

What is the difference in the computation of gradient for gradient descent and jacobian computation for the same cost function. If $y=W^\top x+b$ where $y$ and $x$ are d dimensional and weight of size ...
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1answer
70 views

Least Squares: Derivation of Normal Equations with Chain Rule (Revisited)

My question pertains to someone else's answered question that has made me curious. The OP wanted to differentiate the following using the chain rule: $$ J(\theta)=\frac12(X\theta-y)^T(X\theta - y) $$ ...
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1answer
21 views

How do I calculate the relation between angle parameter($t$) and arc lenth parameter($s$)?

Calculate the flux of the vector field $$\vec F=\left(2\pi x+\frac{2x^2y^2}{\pi} \right) \vec{\imath}+\left(2\pi xy-\frac{4y}{\pi}\right)\vec{\jmath}$$ along the outward normal across the ellipse $x^2+...
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1answer
25 views

Does the Divergence Theorem apply to surfaces with inward-facing normal vectors?

Conceptually I'm having some trouble with understanding the divergence theorem. In the derivations and definitions I'm finding, it always assumes outward facing normals. If I assume inward-facing ...
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29 views

Differential form of a vector equation

I'm reading a text on special relativity, in which we start with an equation: $$\vec u'=\frac{\vec u-\vec v}{1-\vec u.\vec v\ /\ c^2}+\frac{\gamma}{c^2(1+\gamma)}\frac{\vec v\times(\vec v\times\vec u)...
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143 views

Find a closed surface for which $\iint_S \textbf{F} \cdot d\vec{S}$ is negative.

$\text{div}\textbf{F} = x^2+y^2+1$. Find a closed surface for which $$\iint_S \textbf{F} \cdot d\vec{S}$$ is negative or otherwise state why it's not possible. $\textbf{My attempt}$: If $S$ and $D$ ...
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90 views

What is the vector form of an ellipsis' major and minor axis, given lissajous-figur notation for the ellipsis?

A lissajous figure is described by $$ t \rightarrow \begin{pmatrix} A_x \sin(ω_1t + φ_1)\\A_y \sin(ω_2t + φ_2) \end{pmatrix}, t\in \left[0,\infty\right] $$ for $ω_1 = ω_2$ you get a simplified form ...
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5 views

shift and polar

Consider the transformation T: x = \frac{36}{45}u - \frac{27}{45}v, \ \ y = \frac{27}{45}u + \frac{36}{45}v enter image description here
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33 views

Integration by parts of curl in Woltjer’s paper

I am reading A Theorem on Force-Free Magnetic Fields (1958) by L. Woltjer. He used “integrating by parts” twice and I am confused about his results. The first one is in the equation (7). I understand ...
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1answer
72 views

Why is the derivative of tangent vector always along $y$ axis?

Imagine any curve $y=f(x)$ in a cartesian coordinate system. At any point, A vector along the tangent can be given as $$ \vec V = \hat i + \frac{dy} {dx} \hat j $$ I'm trying to find the direction of ...
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21 views

What is the mathematical significance of the integral of the normal derivative?

I found from Advanced Engineering Mathematics by Kreyszig that for the region $R$ and a scalar function $w(x, y)$ the following holds: $$\iint_R \nabla^2 w \ dxdy = \oint _{\partial R} \frac{\...
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1answer
31 views

How is the $\nabla^n $ operator defined?

In Quantum Mechanics, the translation operator $\hat{T}$ can be written as $$\hat{T}(\boldsymbol{x}) = 1 - \dfrac{ix\cdot \hat{p}}{\hbar} - \dfrac{i(x\cdot \hat{p})^2}{2\hbar^2} - \dfrac{i(x\cdot \...
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15 views

How do I calculate the unit vector of P = (3, 4, 5) and Q = (5, 8, 9) (line l going through these two points)?

My textbook says that the answer to this question is: unit vector v = (Q − P) / |Q −P| = (1/3) [1 2 2]. I don't see how they got here, this is my calculation: (Q - P) = [5 - 3, 8 - 4, 9 - 5] = [2 4 4]....
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1answer
29 views

Implicit function theorem, differentiable path

I need to show that the equations: $$x^2y+xy^2+t^2=1$$ $$x^2+y^2 -2yt=0$$ is difinding a differentiable path $\vec{\gamma}=({x}_{(t)},{y}_{(t)})$ at the point $(x,y)=(-1,1)$. after that I should find ...

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