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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12 views

A question about integral transformation from the Unit sphere to Euclidean space.

I want to show the following identity about integral on spherical coordinate system: Let $\mathbb{S}^{d-1}$ denote the unit sphere of the Euclidean space $(\mathbb{R}^d,||\cdot||)$ equipped with the ...
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23 views

On the properties of smooth surfaces

My course material has stated some properties for smooth surfaces and I would like to get some intuition or clarification for these. Let $f = (f_1, . . . , f_{n−m}) : U → \mathbb{R}^ {n−m} $ be a $C^1$...
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11 views

Find the flux $\int_{\gamma} X$ given the drawing

Let $X$ be a vector field $X:\mathbb{R}^2-\{A,B,C\} \to \mathbb{R}^2$, with $curl(X)=0$. Let $\alpha, \beta, \gamma$ be closed curves like in the picture: We know that $\int _{\alpha}X=1$ and $\int_{\...
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Searching a unit square with a point?

For motivation, imagine a pair of analog dials that turn continuously from 0 to 1 - how can you turn those dials in such a search pattern, that eventually every pair of possible dial positions is ...
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2answers
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Gradient of norm of gradient of scalar field

I have a function $f:\mathbb{R}^2\rightarrow\mathbb{R}$ representing a surface $z=f(x,y)$ in euclidean space. Is there an elegant way to apply the chain rule to expand $\nabla||\nabla f||_2$, where $||...
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Differentiability using gradient of a scalar function [closed]

How should I approach this problem? U and V are differentiable functions of x,y and z. Show that a necessary and sufficient condition that u and v are functionally related by equation F(u,v) =0 is ...
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Doubt about notation in Robust Optimization

I'm studying the pricing model under Robust Programming here described from page 41 to page 45. I don't understand what's the meaning of subscripts $i$ and "second" $t$ referred to dual ...
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21 views

Norm of a defined function

I know that for a function $f: R^n \times R^n \longrightarrow R$ \begin{align*} f(x,y ): & = \frac{1}{2} \|x- y\|^2_2 \\ &= 1/2(x^\top x - x^\top y - y^\top x +y^\top y) \end{align*} How ...
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22 views

Gradient of norm squared [closed]

I saw this equation in KKT conditions .Could someone please provide a proof for the following rule: enter image description here How it is gradients of Lagrangian in terms of zn ؟
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Proving spheres are orthogonal

Given two spheres in $\mathbb{R}^3$: $x^2+y^2+z^2=2ax; \ \ \ x^2+y^2+z^2 = 2by$ and $a,b>0$, and $\gamma$ the intersection of the spheres, show that for any $p_0 \in \gamma$, the spheres are ...
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How can I simplify calcul of gradient?

Goal: Given $f(\alpha) :=\dfrac{1}{2} \|y-X\alpha\|_2^2$ I want to show that $\nabla f(\alpha)=0\iff X^T X\alpha = X^T y$ where: $X\in\mathbb{R}^{n\times p},\alpha\in\mathbb{R}^p,y\in\mathbb{R}^n$. If ...
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1answer
42 views

What is wrong with this way of calculating a volume element?

I know that what follows is wrong and I know one needs to use the jacobian to calculate a new volume element $dV$ after a change of variables. However, I don’t understand why the following goes wrong: ...
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26 views

Relationship between Conservative Vector fields and holomorphic functions exerting a primitive

Main thoughts I am currently studying Complex Analysis and I have been noticing the striking resemblance between the requirements for a function to exert a primitive on some open set U and the ...
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Show that the equation $x^2 -ze^{x+y+z} = 0$ defines a surface in the neighborhood of the origin.

Show that the equation $x^2 -ze^{x+y+z} = 0$ defines a surface in the neighborhood of the origin. Letting $f(x,y,z) = x^2-ze^{x+y+z}$ and noting that $\nabla f(0,0,0) =(0,0,-1) \ne 0$. Particularly $\...
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Numerical recipe for interpolating a gradient in n dimensions

I have a large set of points in $n$-dimensional space and each point has a scalar associated with it. I would like to compute the gradient of the associated function. If the points I'm considering ...
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36 views

Which one of the following vector fields is a vector potential for the vector field?

Which one of the following vector fields is a vector potential for the vector field $\mathbf f(\mathbf r) = (z,0,x)$? $\mathbf A = 1/2(x^2+z^2)i$ $\mathbf A = 1/2(x^2+z^2)j$ $\mathbf A = 1/2(x^2-z^2)...
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Conservative Vector Field Theorem

$\underline{\bf{F}}$ is a conservative Vector field $\implies$ $\exists $ a scalar field $\phi$ s.t $\underline{\bf{F}} = \underline\nabla \phi$ How do you show this? I guess you define scalar field $...
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26 views

How do I write a vector as the sum of a vector in two subspaces?

How do I write vector y as the sum of a vector in each subspace? I confirmed that the subspaces are orthogonal and any two vectors out of the 4 are orthogonal to each other. So then I did this: $$\...
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1answer
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Sum of decomposition coefficients of collinear points is $1$, but here seems to be a violation.

Below is a proposition in a text on analytic geometry: Given two points $A$ and $B$, $A\ne B$, $O$ an arbitrary point, if point $C$ is collinear with $A$ and $B$, then vector $\overrightarrow{OC}$ ...
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Gradient / Rotational Vector Fields

What I have done so far: I know that only irrotational vector fields can be gradient fields and only solenoidal vector fields can rotational fields. However I do not know how to check this with these ...
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1answer
43 views

Surface Integral with vector field question?

Let $V$ be a volume in $\mathbb{R}^3$ bounded by a simple closed piecewise-smooth surface $S$ with outward pointing normal vector $\mathbf{n}$. For which one of the following vector fields is the ...
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27 views

Conclusion from Stokes' Theorem

In my book there is a conclusion that if $S$ is a closed surface then $$\oint_Srot\vec{A}\cdot d\vec{\sigma}=0$$ I don't understand why it's always equal to 0.
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Find the distance traveled during the given time interval?

this question confused me I need help can someone help me out please with part b? ill be very pleased and thankful the answer always been wrong have no idea how to solve show work please
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1answer
62 views

Riemann integral in two variables

Let $f:[0,1]\times[0,1]\to \mathbb{R}$, be defined by: \begin{equation} f(x,y)=\begin{cases} 0 & \text{ if }\; 0\leq x<\frac{1}{2},\\ 1 & \text{ if }\; \frac{1}{2}\leq x \leq 1. \...
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1answer
44 views

Vector calculus “identities”

Do these "identities" even make sense? $F$ is a 3D vector field. For the second equation, the LHS is a vector but the RHS is a scalar. $$\nabla\cdot\Delta F=\Delta(\nabla\cdot F),\,\nabla\...
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2answers
28 views

Surface integral with unit sphere

Let S be the surface defined by $$z^2+y^2+x^2=1.$$ Compute the surface integral $$\int_S -z^2-y^2-x^2dS.$$ How can I approach this? I've taken a look at multiple examples, but I don't think I've ...
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1answer
30 views

Proving a version of Green's formula in functional analysis

$\textbf{b}:\Omega\to\mathbb{R}^2$ is a $C^1$ vector field. Need to prove the following version of the Green's formula: $$(\nabla u,\textbf{b}v)=-(u\textbf{b},\nabla v)-(u,v\nabla\cdot \textbf{b})$$ ...
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48 views

How to calculate this second partial derivative?

Let $f\in C^2(\mathbb{R}^2,\mathbb{R})$, $g(\rho,\theta)=(\rho\cos\theta,\rho\sin\theta)$ with $\rho\gt 0$ and $\theta \in [0,2\pi)$. Let $h=f\circ g$, we calculate: $$\nabla h(\rho,\theta)=\nabla f(g(...
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29 views

How is the partial derivative with respect to normal related to the gradient of a scalar field?

I've been trying to understand the solution to a question based on the Gauss Divergence Theorem. The question says, If $\phi$ is harmonic in $V$, then $\iint_{S} \frac{\partial \phi}{\partial n} dS = ...
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30 views

Definition of monotonicity on multivariable functions $f:\mathbb{R}^d \rightarrow \mathbb{R}^d$

I have a question about the monotonicity definition when it's applied to high-dimensional functions $f:\mathbb{R}^d\rightarrow \mathbb{R}^d$. In the answer of my other question: Monotonicity of Convex/...
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19 views

How to find V ,if its curl is given?

Differentiable vector function V such that $$1)\nabla\times \boldsymbol{V} = \boldsymbol{r}$$ $$2)\nabla\times \boldsymbol{V} = \boldsymbol{ 2i + j + 3k}$$ Now I am unable to deduce how to calculate V ...
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2answers
34 views

How should I evaluate the flux?

I have the following vector field $F= \frac{11x}{\left(x^2+y^2+z^2\right)^{3/2}} \vec{i} + \frac{11y}{\left(x^2+y^2+z^2\right)^{3/2}} \vec{j} + \frac{11z}{\left(x^2+y^2+z^2\right)^{3/2}} \vec{k}$ I ...
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8 views

linearity condition of directional derivatives and total derivative

Let $D$ be an open disc in $\mathbb{R}^2$, $a$ be a point in $D$ and $f:D\rightarrow\mathbb{R}$ be a function. Let $u=(u_1,u_2)$ be a non-zero vector. Suppose that (i) the directional derivative of $f$...
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39 views

A question about Percetron Gradient Descent MSE Algorithm.

Let's assume that we have the risk function: $J(a)=\|Ya-b\|^2$. If we take the gradient of this function and claim that is zero, we have: $Y^T(Ya-b)=0$. Now the algorithm is $a(1)$ arbitrary $a(k+1)=a(...
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1answer
24 views

Proof involving derivative product rule for transpose and gradient

Let $f,g: \mathbb{R} \rightarrow \mathbb{R}^n$ be differentiable functions and let $h: \mathbb{R} \rightarrow \mathbb{R}$ be defined by $h(t) = f(t) \cdot g(t)$. Prove that $\frac{dh}{dt}(a) = (Df(a))^...
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1answer
66 views

Value of surface integral with triangle

Let S be the triangle with vertices at the origin and at a pair of vectors a,b $ \in \mathbf R^3 $ with a × b = ( 0, 3, 4) with unit normal vector n pointing in the direction a × b and let f=(1,2,1). ...
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1answer
42 views

Normal Line to a surface

What is the equation of the normal line to the surface r(u,v)=(cosu, sinv+sinu ,cosv) at the point (1,1,0)? What I have done so far: I have found the partial derivatives of the surface separately with ...
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2answers
47 views

How do I calculate an integral of $F$ where $F = \langle y,x^2\rangle$?

I have to calculate the above integral and I'm not sure how to do it. First I find the curl of $\langle y,x^2\rangle$ and I get: $$\langle0,0,2x-1\rangle$$ Therefore it's not conservative because it's ...
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38 views

Solenoidal Fields and Calculating vectors

Check that the vector field f(r) = (z-y,x-z,y-x) is solenoidal, and find a vector a such that f=a*r. what is the vector ||a||^2? What I have done: I have check that the vector field is solenoidal as ...
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1answer
70 views

What is the flux of $\mathbf{f}$ through S along its normal vector?

Let $\mathbf{f}:\mathbb{R}^3\to\mathbb{R}^3$ be a continuous vector field which is parallel to the tangent plane at each point of a piecewise-smooth simple surface S. What is the flux of $\mathbf{f}$ ...
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1answer
38 views

Value of surface Integral

Let S be the part of the surface $z = x^2+y^2$ which lies under the plane $z=4$. What is the value of the surface integral $\iint_S z \, ds \, \,$? What I've done so far: I have used the surface ...
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1answer
42 views

finding $f(x,y)$ that satisfies tangent-plane equation

Q: Give an example for a function $f(x,y)$ continious partial derivatives that satisfies that following: $z=f(x,y)$ is not a plane $z=f(x,y)$ passes through the point $(6,4,3)$ The tangent plane to ...
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1answer
105 views

How do I determine from a picture of a vector field if it's a possible formula for the vector field and conservative or not?

I have an image here of a vector field $F(x,y)$ and am tasked to do the following things: True or false: : A possible formula for $F(x, y)$ is $F(x, y) = <−y, x>$ Is $F$ (the vector field in ...
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1answer
74 views

Finding a vector given a vector field.

Given $\mathbf{f}(\mathbf{r})=(z-y,x-z,y-x)$. Find a vector $\mathbf{a}$ such that $\mathbf{f}=\mathbf{a}\times\mathbf{r}$.
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51 views

Line integral of a rectangle in the $(xy)$-plane with the counterclockwise orientation.

Let $C$ be the boundary of the rectangle with corners $(0,0), (2,0), (2,1), (0,1)$ in the $(xy)$-plane with the counterclockwise orientation. Calculate the line integral: $$ \oint_C (xcos(x^2+y^2)+2y)...
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2answers
81 views

surface integral with normal unit vector

Let $\vec f(\vec r) = (y,-x, zxy)$ and let $S$ be the surface $x^2+y^2+3z^2=1$ , $z≤0$, with unit normal vector $\vec n$ pointing in the positive $z$-direction. The value of the surface integral $$\...
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1answer
53 views

Calculating the area of part of the surface area $x^2+y^2+z^2=4$ which lies inside the surface $x^2+y^2=2x$

The area of the part of the surface $x^2+y^2+z^2=4$ which lies inside the surface $x^2+y^2=2x$ is equal to $n(\pi-2)$ for an integer n. What is the value of n? What I know so far: I get that my answer ...
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1answer
114 views

Differentiable scalar fields question.

Let $f,g:\mathbb{R}^3\to\mathbb{R}$ be twice continuously differentiable scalar fields. Which of the following statements is false? A: If $S$ is any simple piecewise-smooth surface with unit normal ...
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0answers
21 views

The calculation of $\nabla_{x_i} \bigg(1-\frac{d_{ij}^2}{\|x_i-x_j\|^4}\bigg)$, $x_i, \, x_j \in \mathbb{R}^2$

Let $x_i,\,x_j \in \mathbb{R}^2$. I am trying to calculate $$\nabla_{x_i} \bigg(1-\frac{d_{ij}^2}{\|x_i-x_j\|^4}\bigg)$$ My work is the following \begin{align*} &\nabla_{x_i} \bigg(1-\frac{d_{ij}^...
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0answers
6 views

Path dependence in the presence of singularities

There is a Field that is categorized by {m[x, y], n[x, y]} = {-(y/(x^2 + y^2)), x/(x^2 + y^2)} As we can observe from above the field has singularities on {0,0}. ...

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