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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1answer
29 views

Hessian of a squared bilinear form

I have to expand the function $f:\mathbb{R}^n \times \mathbb{R}^m \rightarrow \mathbb{R}$ in Taylor series where $$ f(x,y) = (x^TAy + B^Tx + C^Ty)^2$$ with $A\in\mathbb{R}^{n\times m}$, $B\in\mathbb{R}...
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7 views

Further conditions on a solution to a homogeneous linear system

Given a homogeneous linear system given by a positive semi-definite matrix A: $$Ax = 0,$$ I would like to know the most sensible way to also involve the extra constraint $$\sum_i x_i \neq 0$$ and ...
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1answer
19 views

Gradient points in the direction of deepest ascent

this could be an old question to many of you but very new to me, As I got from some theory, gradient points to the direction that helps a function f(x) increases (or ascends to a higher value)? That ...
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1answer
39 views

Calculating path of a particle in a vector field

I have a file with .dat extension in which I have data for a vector field in form of columns arranged in order {position vector_x, position vector_y,x_magnitude,y_magnitude}. If I pick a specific ...
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18 views

Gradient vs Directional Derivative Visually

Gradient vs Directional Derivative As you can see in the image I have shown Gradient of hill by blue vector. Vector red indicates direction of max slope. I have understood the gradient. The problem I ...
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18 views

Linearised Euler equations in rotating frame

The Euler momentum equation for an inviscid fluid of constant density $\rho$ is $\rho\left[\frac{Du}{Dt} + \underbrace{2\Omega\times u}_{\text{Coriolis }} + \underbrace{\Omega\times (\Omega \times x)}...
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1answer
25 views

About the conservative and not conservative vector field

I was wondering why (3) is conservative while (4) is not. I thought a conservative vector field should be a closed loop, and the two vector fields should have similar results since they both excluded ...
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0answers
41 views

If two functions are equal, why is it not a contradiction for one to be at a minimum while another is at a maximum?

“Consider the PDE $u=u_{xx}+u_{yy}$ in dimension 2, where $u:\mathbb{D}\to\mathbb{R}$ is a scalar function on the unit disk $\mathbb{D}=\{(x,y)|x^2+y^2\leq1\}$. Show that if $u$ is a solution that is ...
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1answer
34 views

Line integral of a Vector Field around a Closed path

This is just a little question. Suppose you want to evaluate an integral around a closed path formed by a curve $C(t) $(only one curve), I suspect that the result would be $0$, because you will do an ...
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1answer
60 views

Proof of Vector Triple Product by Directions and Magnitudes [duplicate]

I'm trying to prove the vector triple product expansion by magnitude and direction: $$ \vec{a} \times(\vec{b}\times \vec{c})=(\vec a\cdot \vec{c})\vec{b} - (\vec{a} \cdot \vec{b})\vec{c} $$ The ...
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2answers
34 views

Line integral counterclockwise.

let $F$ be a vector field, such that $$F(x,y)=\langle x^2y^2,xy^2\rangle$$ Calculate the integral of $F$ along the closed formed by the line $x=1$ and the parabola $y^2=x$. first, we have to ...
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1answer
71 views

Does Gauss' theorem and Stokes' theorem still hold for distributions?

We can define the derivative of distributions and then we can define their grad, div and curl, so do the Stokes' theorem and Gauss' theorem still hold for distributions? if not, under which condition ...
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21 views

Scalar-by-vector derivative involving L2 norm and Hadamard product

I have a function $f(\mathbf{x})$: $\mathbb{R}^N \rightarrow \mathbb{R}$ given by: $f(\mathbf{x}) = \lvert \mathbf{A}(\mathbf{x} \circ \mathbf{x})-\mathbf{b} \rvert^2$. with $\mathbf{A} \in \mathbb{R}^...
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1answer
31 views

Solving linear systems by preconditioned conjugate gradient algorithm

Let a linear system, AU=F, be arisen from the second order central finite difference approximation to a Poisson boundary value problem as follows $$−\Delta u = f \in \ \Omega := (0,1) \times (0,1)$$ ...
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1answer
30 views

Finding the steepest descent (gradient descent)

My question is regarding question 2c in the attached image: Luke is at the beacon, What direction should Luke head to cool off the fastest? The answer provided to me was $\langle -2e^4, 4e^4, -2e^4 \...
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43 views

How to generalize the various vector calculus theorems to distributions?

https://en.wikipedia.org/wiki/Vector_calculus_identities Here are the list of vector calculus identities, in the proof of these identities, we all assume that these functions are $C^k$ in an open set, ...
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11 views

Curl of cross product of two vector-valued functions

In Madsen and Tornehave's From Calculus to Cohomology Page 4, there's a formula which the text says can be obtained by straightforward calculations but I don't know how, I hoped to use vector calculus ...
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1answer
12 views

Can we obtain the one parameter function that represents the arguments progression on gradient progression of a multiple parameters function?

Assume that $(a_1,a_2,...,a_n) \in \Bbb R^n$ and the $F: \Bbb R^n \rightarrow \Bbb R$ $(x_1,x_2,...,x_n) \longmapsto F(x_1,x_2,...,x_n)$ is differentiable function at all parameters. Is there any math ...
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1answer
27 views

Stokes' Theorem - cylindrical coordinates

I'm currently having an issue with verifying the validity of Stokes' Theorem on a particular problem. I can solve the problem by using Stokes' theorem to turn a surface integral of the curl of a ...
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1answer
23 views

Taylor expansion of $\frac {1}{|x-y|}$with x and y two vectors

This equation comes from a physics script on electrodynamics, saying that this equation comes from a Taylor series expansion. I understand the first equality, but not the second one. It is really not ...
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1answer
15 views

To find the angle a particle makes with the horizontal at any time 't'

Should you vector sum the position vectors at this time 't' or vector sum the velocity vector at this time 't' to find the angle a particle makes with the horizontal at any time 't'
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11 views

How to solve streamline and its curvature

Consider the following vector field F in Cartesian space (x,y,z) $$ F_0(x,y,z) = -\cfrac{1}{(x^2+y^2)}(x+y, y-x, z) $$ What is the curvature of its streamline? I've tried to solve by: $\cfrac{dx}{-\...
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2answers
39 views

Finding Potential for a conservative vector field using Line Integral

The vector field $\textbf{A} = (yz , xz , xy ) $ in cartesian coordinates is irrotational, or has zero curl. This means there should exist a function $\phi : \mathbb{R}^3 \rightarrow \mathbb{R} $ with ...
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1answer
58 views

Find potential function of a vector field

We have $\vec{F}(x,y,z) = (2xcos(y)-2z^3)i + (3+2ye^z-x^2sin(y)) j + (y^2e^z-6xz^2) k$ Now a potential field $\vec{F}(x,y,z) = grad(\phi)(x,y,z) = f_x(x,y,z)i+f_y(x,y,z)j+f_z(x,y,z)k$ But after taking ...
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24 views

Curl of a Laplacian

Suppose I have the following function: $$A\Delta\underline{U}=0$$ where $A$ is a constant, $\underline{U} = U_x\underline{e}_x + U_y\underline{e}_y$ is a bidimensional first order tensor, and $\Delta$ ...
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1answer
43 views

About the formula for the magnitude of the gradient of a scalar valued function, $|\nabla T|$

Let $T: R^3 \rightarrow R$. I recognize that $|\nabla T|$ is the rate of change of $T$ in the direction of the most rapid increase. Say we move by $d\vec{l} = (dx, dy, dz)$. When $d\vec{l}$ is ...
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1answer
20 views

Let $T$ be a triangular region in $\mathbb{R}^2$ defined by $x\geq 0, y\geq 0, x+y\leq 1$:

Let $T$ be a triangular region in $\mathbb{R}^2$ defined by $x\geq 0, y\geq 0, x+y\leq 1$: a) Describe the boundary of $T$ b) Prove that the point $(0.0001, 0.9998)$ is an interior point of $T$ ...
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2answers
65 views

Integral over a closed area

I consider a closed area $ A $ and a piece of area $dA$ from it. Now let be $ \underline{n} $ the normal vector of $ dA $. Then we call $ d\underline{A}:=\underline{n}\cdot dA $ the area vector. Now ...
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41 views

Mean Value Theorem in a Inner Product Form

$f: U \to \mathbb{R}$, is strongly convex and twice differentiable, where $U$ is a convex set and $U \subset \mathbb{R}^d$. There are two points, $x_0$ and $x_1$ both in $U$. In particular, $\nabla_x ...
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1answer
57 views

Proving $(\nabla \times \mathbf{v}) \cdot \mathbf{c} = \nabla \cdot (\mathbf{v} \times \mathbf{c})$ using cylindrical coordinates

Assuming the form of divergence in polar coordinates is known, I am attempting to use the following definition of the curl of a vector field to determine the form of the curl in cylindrical ...
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0answers
52 views

Relation between a scalar function and its “inverse” gradient

I have a surface defined by the implicit formula: $$ F(\boldsymbol{x}) = 0 $$ where $ \boldsymbol{x} \in \mathbb{R}^n$ and $F : \mathbb{R}^n \to \mathbb{R} $ (actually I have $n=3$, but if the result ...
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0answers
26 views
+50

The proof of the multivariable chain rule (Vector form)

Let $f$ be a function defined on an open set $U$, and let $X(t)$ be a curve such that $X(t)$ is contained in $U$ for all $t$. We define the function $g(t)=f(X(t))$, I want to know what is $dg/dt$, ...
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1answer
19 views

What is the relationship between $\mathbb{E}(\|\mathbf{X}\|)$ and $\|\mathbf{Y}\|$?

Given that we have two vectors $\mathbf{X}\in\mathbb{R}^N$ and $\mathbf{Y}\in\mathbb{R}^N$, where $\mathbf{X}$ is a random variable with $\mathbb{E}(\mathbf{X}) = \mathbf{Y}$. Here $\mathbb{E}$ ...
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1answer
29 views

Differential of multi variable function

I'm trying to find the differential of the following multi variable function and then use the external definition of gradient in order to find its gradient: $$ f(\overline{x})=\phi(A\bar{x}) \,,where\,...
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1answer
48 views

Solving vector calculation question

A spaceship is traveling through space at constant speed along a straight line that passes through the points $A = (-3,-8,-6)$ and $B = (-8,-4,-3)$. The star Gliese $061$, located at $P = (-12,0,0)$ ...
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0answers
35 views

Munkres, Analysis on Manifolds, Problem 38-5

I am having trouble doing this problem. I know that we need to use Stokes' Theorem. $D_3$ is easy. The problem is I can't seem to picture a shape whose boundary is $D_1$ and $C_1$ and/or $D_1$ and $...
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2answers
21 views

Multivariable divergence theorem work

Find the flux of the vector field $\vec{G}=\operatorname{curl} \vec{F}$, where $$ \vec{F}(x, y, z)=y^{3} \vec{i}+x^{3} \vec{j}+z^{3} \vec{k} $$ through the upward oriented part of paraboloid $$ z=1-x^{...
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0answers
29 views

$f(x,y,z)=0$, and $z_{xx}z_{yy} -z^2_{yx}=0$, then $x_{yy}x_{zz} -x^2_{yz}=0$ too.

Please consider $f(x,y,z)=0$, such that $f$ is a $C^{\infty}$ function in the neighborhood of $p=(p_1,p_2,p_3)$ on $\mathbb{R}^3$, and $f(p)=0$. Also assume that $$ \dfrac{\partial f}{\partial x}(p) \...
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0answers
31 views

isn't it possible to get partial derivatives for x^(1/3)y^(1/3)?

I'm studying "vector calculus written by Jerrold E.marsden, Anthony Tromba 6th Ed" My problem is I couldn't understand example 4 in 2.3section They said that they can get partial derivatives ...
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1answer
26 views

Multivariable divergence theorem bicylinder questiin [closed]

Let $E$ be the intersection of the cylinders $x^{2}+y^{2} \leq 1, y^{2}+z^{2} \leq 1$. Compute flux $ \ \iint_{\partial E} F \cdot d S$ where $\vec F = \left(x y^{2}+\cos (y z)\right) \hat i - \left(x^...
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0answers
25 views

Multivariable divergence theorem two cylinder question [duplicate]

Let $E$ be the intersection of the cylinders $x^{2}+y^{2} \leq 1, y^{2}+z^{2} \leq 1$. Compute the flux $$ \begin{align} & \iint_{\partial E} F \cdot d S \\[6pt] \text {where } & F=\left(x y^2+...
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1answer
39 views

Let $f(X)=\|X\|$ and $g(X)=\|X\|^2$, $X\in \mathbb{R}^n$.

Let $f(X)=\|X\|$ and $g(X)=\|X\|^2$, $X\in \mathbb{R}^n$. Give an example of points $X$ and $Y$ at distance one unit, such that. $a)$ $|g(Y) - g(X)|> 10^{60}$ $b)$ Show that $f$ is uniformly ...
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1answer
50 views

Use the chain's rule to find $f'(x,y,z)$ if $f(x,y,z)=\int_{\sin x}^{yz} g(t)dt$

Let $g:\mathbb{R} \rightarrow \mathbb{R}$ continuous and $f:\mathbb{R}^{3} \rightarrow \mathbb{R}$ defined for $f(x,y,z)=\int_{\sin x}^{yz} g(t)dt$. Express $f$ like the composition of differentiable ...
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3answers
52 views

Evaluate the line integral $\int_\gamma \frac{-y}{x^2+y^2}dx + \frac{x}{x^2+y^2}dy + \ln{(z^4+z^2+1)}dz$ using Stokes' Theorem

I found the following problem in a textbook (translated): Evaluate $$\int_\gamma \frac{-y}{x^2+y^2}dx + \frac{x}{x^2+y^2}dy + \ln{(z^4+z^2+1)}dz$$ where $\;\gamma\;$ is given by the intersection ...
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1answer
14 views

A vector-product formula

Let $\mathbf a, \mathbf b,$ be vectors in $\mathbb R^3$ and let $R$ be a $3\times 3$ matrix. Then we have $$ ^t\!R\bigl(R\mathbf a\times R\mathbf b\bigr)=(\det R)(\mathbf a\times \mathbf b). \tag 1$$ ...
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0answers
36 views

Is there a way to unify curl, gradient, and divergence operators as special cases of a more general operator?

I realize there's probably an answer to this question somewhere on this site, but it would seem I'm having trouble picking the right search terms. In my multivariable calculus class, the professor ...
2
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2answers
59 views

Vena contracta effect, why can't streamlines change direction abruptly?

I am curious about the common explanation for the vena contracta effect that occurs as a flow moves around a sharp corner, or within a free jet of liquid issuing from a nozzle. The explanation goes ...
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0answers
29 views

Trouble understanding proof for Lagrange multipliers

I'm working through the proof on Lagrange multipliers in Multivariate Calculus and Geometry by Sean Dineen. For, reference, these are the pages which contain the proof. Page 1 Page 2 My confusion ...
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0answers
20 views

Describe the level surfaces and a section of the graph of each function:

Describe the level surfaces and a section of the graph of each function: $a)$ $ f: \mathbb {R} ^ 3 \rightarrow \mathbb {R}, \: (x, \: y, \: z) \mapsto -x ^ 2-y ^ 2-z ^ 2 $ $b)$ $ f: \mathbb {R} ^ 3 \...
4
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1answer
55 views

Evaluate the surface integral $\iint_\Sigma (y^2-z^2)e^{yz}\,ds$ using Stokes' Theorem

I found the following problem in a textbook: Evaluate $$\iint_\Sigma (y^2-z^2)e^{yz}\,ds\,,$$ where $\;\Sigma\;$ is given by $x^2+y^2+z^2=1$, $\;z\geq0$, by evaluating the rotation of the field $\;\...

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