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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-3
votes
3answers
57 views

Linear Algebra with basis [closed]

Consider the following set: $$S=\{(1+i,i,0),(1-i,1,0),(2,i,-i)\}$$ Is it a basis of $\mathbb{C}^3$?
0
votes
1answer
15 views

Finding the radii of an ellipse from the intersection of a plane and a sphere

I'm trying to solve the following problem, regarding Stokes Theorem: $F = z i + xj + yk $; C the curve of intersection of the plane $x + y + z = 0$ and the sphere $x^2 + y^2 + z^2 = 1$ [Hint: ...
4
votes
0answers
19 views

Poincaré-Hopf Index Theorem - Intuitive explanation

I heard about an interesting theorem. Poincaré-Hopf Index Theorem : If $\vec{v}$ is a smooth vector filed on the compact, oriented manifold $X$ with only finitely many zeros, then the global ...
1
vote
2answers
97 views

Finding extrema points with lagrange multipliers

Using lagrange multipliers, find all the extrema points of the function $f(x,y) = x^2 + (y-b)^2$ subject to the constraint $y = x^2$. Using the fact that critical points occur at $\triangledown ...
1
vote
1answer
66 views

Kinetic energy of incompressiblue fluid

I am trying to show that the kinetic energy for an incompressible and irrotational fluid with no sources and no sinks is given by $$\frac{\delta}{2} \iint_{S} \psi \frac{\partial \psi}{\partial n} ...
3
votes
1answer
32 views

Confused on surface integral problem

I am asked to evaluate $$\iint_{S} [ \nabla \phi \times \nabla \psi] \bullet n dS$$ where $\phi=(x+y+z)^2$ and $\psi=x^2-y^2+z^2$ where S is the curved surface of the hemisphere $x^2+y^2+z^2=1$ , $z ...
1
vote
1answer
21 views

question on vector calculus notation

I just have a question about the vector calculus notation: $$(u \cdot \nabla)u$$ Is that the same as $( \nabla \cdot u)u$?
0
votes
0answers
15 views

Circulation around a curve

in a previous question I had asked How to apply the divergence thereom in the plane About the divergence of $F=(xy)i+(2x-y)j$ where C is the triangle with verticies $(0,0)$ , $(1,0)$ and $(0,1)$. ...
0
votes
1answer
34 views

Gradient of curvature over triangle

In an article "Smooth Feature Lines on Surface Meshes", there is this in Equation (4): ∇Ki(T), i = {min, max} where K is principal curvature and T is a triangle. How do I calculate this?
0
votes
1answer
36 views

vector calculus using Lagrange Multipliers

$(1)$ Let $c\in R$ be a constant. Using Lagrange Multipliers, find all the extrema of $$f(x,y) = x^2 + (y-c)^2$$ subject to the constraint $$y = x^2$$ I'm pretty sure I've found the critical ...
0
votes
1answer
9 views

Vector field line integral: confusion about sign of dl, order of limits

I have some confusion about simple line integrals of vector fields. If I want to calculate integral $\int_C\underline F\cdot d \underline l$ from point $(1,0)$ to $(0,0)$ in a straight line, then ...
0
votes
1answer
13 views

How to show this vector cross product/gradient result

One of my books has that if $$\bar A= \phi \nabla \psi$$ then $$\nabla \times \bar A = \nabla \phi \times \nabla \psi$$ But I don't see why it is true. What is the proof of this? Thanks
1
vote
1answer
30 views

Line integral using Green's Theorem considering equation of ellipse

I am given the integral $$I=\int_{C} \frac{y}{4x^2+7y^2} dx - \frac{x}{4x^2+7y^2}dy$$ where C is the rectangle with vertices $$A=(4, 7), B=(-4, 7), C=(-4, -7), D=(4, -7)$$ oriented in the ...
0
votes
1answer
15 views

Vector difference norm bound of $\|\frac{x_1}{\|x_1\|^2} - \frac{x_2}{\|x_2\|^2}\|$

Given $\|x_1 - x_2 \| \leq C$ where C is a constant, could we derive a bound of $\|\frac{x_1}{\|x_1\|^2} - \frac{x_2}{\|x_2\|^2}\|$?
1
vote
1answer
35 views

Evaluating Line Integral with Green's Theorem

I'm given a line integral $$\int_{C} \left(\frac{\sin(3x)}{x^2+1}-6x^2y\right) dx + \left(6xy^2+\arctan\left(\frac{y}{7}\right)\right) dy$$ where C is the circle $$x^2+y^2=8$$ oriented in the counter ...
0
votes
1answer
26 views

lim r(t) = L implies lim ||r(t)|| = ||L||

The question is: if $\lim_{t\rightarrow t_0}\vec r(t) = \vec L$ show that $\lim_{t\rightarrow t_0} \|\vec r(t)\| = \|\vec L\|$ So here's where I am so far: let $\vec r(t) = (f_1, f_2,...f_n)$ be ...
1
vote
3answers
31 views

Work Done when more than one field exist

Suppose we have two different electric field, $\vec{E_1}$ and $\vec{E_2}$ where $\vec{E_i}$ are elements of $\mathbb R^2$ $y>0 => \vec{E}$=$\vec{E_1} $ and $y<0 => \vec{E}$=$\vec{E_2} $ ...
2
votes
0answers
31 views

Green's theorem with unit normal and del operator

By appropriately choosing the functions P and Q in Green's theorem, show that $\iint_R\nabla^2 \phi\;dA =\int {\partial \phi}{\partial n} \;ds $, where $\frac{\partial}{\partial n}$ denotes ...
0
votes
0answers
13 views

Relating Integration of directional derivative and the original function

Working in a manifold $M = \mathbb{R}^n$, Let $x_i$ be the i-th coordinate function, and $\varphi$ be in the set of germs of all the real valued functions. i.e. $(M,p)\rightarrow R$ and ...
0
votes
1answer
106 views

How to evaluate circulation

I am having some trouble with the following question: Solve for the circulation in five different ways for the velocity field $$\bar v=(e^{-x^2}-yz)\hat i+(e^{-y^2}-xz+2x)\hat j+(e^{-z^2})\hat k$$ ...
1
vote
1answer
33 views

Differentiation that involves determinant and vectors

How can i calculate following? $\frac{d}{dR'} {(|\vec{R}-\vec{R'}|)^{-1}}$ I tried this but I'm not sure this a valid method: let $\vec{R}-\vec{R'} = \vec{u}$, then $-\vec{dR'}= \vec{du}$ ...
1
vote
0answers
18 views

How could I determine the form of a function that chases another function?

This is a problem that a teacher told me about that's been bothering me for a while. I'm positive that this has been explored before because it seems way too useful for physicists to not have come up ...
1
vote
1answer
17 views

Neumann condition for Poisson equation

Solving $ \nabla^2u = 1 $ for spherically symmetric u in the region $r < a, a > 0$, with the following conditions at r = a (separately) (a) $u = 0$ (b) $\nabla u \cdot n = 0 $ where n is the ...
0
votes
1answer
40 views

How to apply the divergence thereom in the plane

I am trying to work a seemingly simple practice problem but I am having some confusion. The question asked to verify the divergence theorem in the plane for the vector field $$F=(xy)i+(2x-y)j,$$ where ...
0
votes
0answers
15 views

Global Clebsch potentials

For an aribitrary vector field $\mathbf{v}$ on $\mathbb{R}^3$, it always can locally be written as $$ \mathbf{v}=\nabla f+g\nabla{h} $$ where $f$, $g$, $h$ are called Clebsch potentials. My question ...
2
votes
1answer
26 views

Trouble plotting Maple space curve given a parametrization

I am being asked to plot a curve C with parametrization $$ r(t)=\left \langle \sin(mt)\cos(nt), \sin(mt)\sin(nt), \cos(mt) \right \rangle $$ with parameters of $$0\leq t\leq 2\pi$$ with integers ...
1
vote
1answer
24 views

Find the volume of ice cream cone using cylindrical/spherical coordinates

I'm stuck on what the boundaries are for the volume bounded by the cone $z=-\sqrt{(x^2+y^2)}$ and the surface $z=-\sqrt{(9-x^2-y^2)}$ $\,\,$-essentially an upside down ice cream cone Remember that ...
0
votes
0answers
20 views

Determinant of Jacobian matrix in coordinate transformation

I have not completely understand a thing. To change coordinate in double integral we need to multiply by the the absolute value of Jacobian determinant. I studied the theorem which rules the he ...
1
vote
0answers
32 views

Confusion with vector addition

So while reading a solid-state physics book, I encountered the following (drift-diffusion equation): I know the divergence of a vector field produces a scalar field, and so the second equation is ...
0
votes
0answers
32 views

Solving for $G(x,y)$ in a Gradient System (Differential Equations)

If I'm given the following, how would I solve for $G(x,y)$? $$\begin{align}\frac{dx}{dt}&=y^2-\cos x\\ \frac{dy}{dt}&=2xy-\sin y\end{align}$$ I know $x'$ is equal to the partial of $G$ with ...
0
votes
1answer
26 views

Change order of integration

I'm stuck on how to change the order of integration for this question, $$\int_0^{2\pi} \int_{\cos x}^1\, f(x,y)\,\,\,dydx $$ We know that if we take vertical strips of the original integral the ...
0
votes
0answers
18 views

Find constants so that the directional derivative of $f(x,y,z) = axy^2+byz+cx^3z^2$ has maximum value $32$ in point $P$ given the direction

I am asked to find $a$, $b$ and $c$ so that the directional derivative of $$f(x,y,z) = axy^2+byz+cx^3z^2$$ has maximum value of $32$ in the point $P(1,2,-1)$ and in the direction $\overrightarrow{u} = ...
0
votes
2answers
41 views

Express the vector field in polar coordinates

$\underline{G}=(8r^7x-5r^3y)\underline{i}+(-8r^7y+5r^3x)\underline{j}$ where $r=(x^2+y^2)^\frac{1}{2}$ How would I express the vector field as cylindrical coordinates? I have looked at various ...
0
votes
1answer
34 views

How can I solve this problem? [closed]

A rotation φ1 + φ2 about the z-axis is carried out as two successive rotations φ1 and φ2, each about the z-axis. Use the matrix representation of the rotations to derive the trigonometric identities
2
votes
0answers
22 views

Solving A System Of ODE's On MAPLE 17

I Have the velocity fields for two vortices that are located at two different points ${\bf{x_1}}(x_1,y_1)$, ${\bf{x_2}}(x_2,y_2)$ $\vec{V_1} = (u_1,v_1) = \frac{\Gamma_1}{2\pi}\frac{1}{(x-x_1)^2 + ...
-1
votes
0answers
11 views

Vector analysis help - Regulated

If a surface is defined as regulated what does this mean and can I have an explained example please. Thanks
1
vote
1answer
24 views

Evaluation of $\int_C{F.dr}$

For a given vector field and curve C, evaluate $\int_C \mathbf{F.dr}$ $\mathbf{F}(x,y)= -y \mathbf{i} +x \mathbf{j}$, C:Ellipse $\frac{x^2}{4}+\frac{y^2}{9}$=1 First we parametrise $x=2\cos{t}$; ...
2
votes
3answers
79 views

Show that there is no vector $e$ such that $e × ~x = ~x$ for all $x$?

(I apologize for not putting an arrow over my vectors, as I couldn't figure out how to type them) Basically I'm trying to show that vectors do not have a multiplicative identity. But I can't find a ...
-1
votes
0answers
16 views

Parameterize path in terms of increasing $t$

Parametrize the path from $\left(\frac{3}{\sqrt{2}},3,9\right)$ to $\left(0,3,0\right)$, along the intersection of the surface $z=2x^2$ and the plane $y=3$ in terms of the parameter $t$, with $t$ ...
1
vote
1answer
43 views

Contour integral involving Riemann sum

$\int_C{xdy-ydx}$; where $C$ is the curve $x=a\cos^3 t$, $y=a\cos^3t$ When I do whatever work for the working,i.e replacing the x,y,dx and dy in terms of t, I get $3a^2\int{(\cos^2t \sin^2t})dt$. ...
0
votes
0answers
14 views

Vector cross product in non-orthogonal basis system

I'm given a non-orthogonal basis system $\vec{u}, \vec{v}, \vec{t}$ and i need to express linearly the following vector $\vec{u} \times \vec{v}$ as a linear combination of the basis vectors. So ...
1
vote
1answer
79 views

Where am I making mistakes in verifying the divergence thereom

I am just looking for anyone to please possibly see where I could be making the mistake. If it is a correct answer of $48 \pi$, why do I keep evaluating the triple integral to be more then it I am ...
0
votes
0answers
10 views

Finding the Stationary Solution of an Advection-Diffusion PDE

Probably this is a trivial question. I am trying to find stationary solutions (steady state solution) of $$ \frac{\partial u}{\partial t} = \nabla. (f(x) u) - D \nabla^2 u $$ where $f(x)$ is a ...
1
vote
0answers
30 views

Biot-Savart Law to construct vector potential for divergence free vector field on $\mathbb{R}^3$

I would like to confirm a method I am trying to use which uses the Biot-Savart Law to construct a vector potential $\underline{w}$ for a divergence free vector field $\underline{v}$ on $\mathbb{R}^3$. ...
1
vote
1answer
27 views

Euclidean norm on integer lattice

Does the Euclidean $ L^2 $ norm (and distance) make any sense on an integer lattice in $ \mathbb{R}^n $? And what is the preferable way of calculating a type of norm in such spaces?
1
vote
1answer
53 views

Gradient of Function

I am trying to find the $\nabla F$ with respect to $x_i$ where $F$ is as follows: $$F(x_0,...,x_n) = c_1\sum_{i=0}^{n}\sum_{j=1}^{k}\frac{1}{||x_i - r_j||_2^2} + ...
0
votes
0answers
21 views

A continuous, path-independent, non-conservative vector field

I am trying to define a non-conservative vector field on $\mathbb{R}^2$ whose line integrals don't depend on its paths. I think it could be $$ F(x,y) = \begin{cases} (x+y, x-y) && \mbox{if ...
4
votes
0answers
89 views

Uniqueness of a PDE solution

Suppose $F:U \to \mathbb{R}^n$ is a $\mathcal{C}^1$ vector field on an open set $U \subseteq \mathbb{R}^n$. Let $\lambda \in \mathbb{C}$. Consider the partial differential equation $$F(x) \cdot ...
0
votes
0answers
36 views

Verification of divergence theorem

At time t the velocity field of a fluid is given by $\bar V(x,y,z)=x^3 {\bf i}+y^3{\bf j}+z{\bf k}$ the outward flux integral $ Φ = \iint_S \bar{V}\cdot d\bar{S} $ where S is the surface of the ...
3
votes
0answers
20 views

solving linear gradient PDE

Let $f:\mathbb R^n \to \mathbb R$ be a function that satisfies the equation: $$\nabla^T f = - (\nabla^T \phi ) f $$ where $\phi:\mathbb R^n \to \mathbb R$ is some given function and $\nabla^T$ is the ...