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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25 views

Work = line integral over closed loop

For a velocity field $$ \textbf G(x, y) = (3x^2 − 6y^2 + 1)\textbf i + (x + 4y − 12xy)\textbf j $$ show that the work done in moving a particle on the unit circle centred at (1, 0) taking an ...
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0answers
18 views

direct and the inverse transformation of basis vector? [on hold]

the relations b/w the direct and inverse transformation .consider two basis $e_1,e_2,e_3$ and $e'_1,e'_2,e'_3$ drawn from the same point $O$.they can be espressed as given in the picture below (from ...
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0answers
26 views

Using Euler's equation

An incompressible inviscid fluid, under the influence of gravity, has the velocity field $$\textbf u = (− \cos(x)\sin(y), \, \, \sin(x)\cos(y), \, \, 0)$$ with the $z$-axis vertically upwards, where $g$ ...
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1answer
20 views

Examples of physical motivation for integrals over scalar field?

I'm looking for good examples of physical motivation for integrals over scalar field. Here is an example I've found (source): A rescue team follows a path in a danger area where for each position ...
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1answer
24 views

Differentiating a function composition

Given $g:R^n \rightarrow R^k$ and $h:R^k \rightarrow R$, we have $f(x) = h(g(x))$. Using the chain rule, we can differentiate $f(x)$ to get $f'(x) = \nabla^Th(g(x))g'(x)$ My question is why do we ...
2
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1answer
21 views

Show that the vector field $\vec F=(yf(u),xg(u))$ has no potential

I'm trying to prove that the vector field $\vec F=(xf(u),xg(u))$ with $u=xy$ is not conservative. I suppose that there is a function $\phi$ so that $\nabla \phi= \vec F$. So I need to satisfy that: ...
3
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1answer
19 views

Show that the vector field $\vec F=(xf(u),xg(u))$ is not conservative

I'm trying to prove that the vector field $\vec F=(xf(u),xg(u))$ with $u=xy$ is not conservative. I suppose that there is a function $\phi$ so that $\nabla \phi= \vec F$. So I need to satisfy that: ...
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1answer
20 views

Calculate the vector surface integral

Let $V=\{(x,y,z)\in \mathbb{R}:\frac{1}{4}\le x^2+y^2+z^2\le1\}$ and $\vec{F}=\frac{x\hat{i}+y\hat{j}+z\hat{k}}{(x^2+y^2+z^2)^2}$ for $(x,y,z)\in V$. Let $\hat{n}$ denote the outward unit normal ...
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0answers
30 views

Physical difference between $\nabla^\bot\cdot u=0$, and $\nabla\cdot u^\bot=0$ and the existence of a scalar potential

If there exists a $2D$ vector field $u=u(x)=(u_1,u_2)$ such that $\nabla\cdot u=0$ is it equivalent to saying following? $$\nabla\cdot u=?(\nabla\cdot u)^\bot=\nabla^\bot\cdot u^\bot=\nabla\times ...
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50 views

Deriving Gradient [closed]

So I'm trying to derive a gradient of the following problem: The negative log-likelihood can be given as: $NLL\left ( \mathit{w} \right ) = -\sum_{i=1}^{N} \left [ \left ( 1 - y_i \right ) ...
2
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0answers
9 views

Vector equation of line containing point and perpendicular to plane [duplicate]

How would one find the vector equation of the line that contains the point (x0, y0, z0) and is perpendicular to the plane Ax + By + Cz = D?
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1answer
49 views

Material Derivative of the Gradient of a Scalar Field

Let $f$ be a scalar field that is continuous and does not vary along the flow, that is $D_t(f)=0$ where $D_t=\partial_t+\vec u\cdot\nabla$ where $\vec u$ is the incompressible velocity field (i.e ...
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1answer
23 views

Rotation matrix according vector

I am stuck on the following two questions. I find formulas for the computation of 3D rotation matrix, but still cannot get how to do those questions. Find matrix for rotation $R_{\theta \bar n}$, ...
1
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1answer
79 views

A thief and a policeman [closed]

A policeman desperately tries to catch a thief that is $a$ meters away. The thief has the constant velocity $v$, and the policeman has the constant velocity $k\cdot v$, with $k > 1$. The policeman ...
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0answers
11 views

A $C^2$ $f$ such that for every $x \in \mathbb R^n$, $t \in \mathbb R$, $f(tx)=t^2f(x)$. [duplicate]

I am trying to do the following exercise: Suppose $f:\mathbb R^m \rightarrow \mathbb R^n$ is $C^2$ and for every $x \in \mathbb R^n$, $t \in \mathbb R$, $f(tx)=t^2f(x)$. Show that there exists a ...
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0answers
27 views

tensor identity for cross product

I've read somewhere the following identity for a tensor rank 2 $ \nabla \times \nabla v =0 $ where $v$ is a vector of "j" components and $\nabla = \frac{\partial}{\partial x_i}$, such that $ \nabla ...
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0answers
28 views

Vector formula for the distance from a point to a line

I am seeking a proof that the distance from a point $\,\mathbf a\,$ to the line joining points $\,\mathbf b\,$ and $\,\mathbf c\,$ is given by $$\frac {|\mathbf a \times \mathbf b + \mathbf b \times ...
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4answers
59 views

Textbook for Vector Calculus

Can anyone recommend a textbook for studying vector calculus (vector analysis) only, that focuses on the theoretical mathematics behind vector calculus? Currently, I am using vector analysis by ...
2
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1answer
18 views

Closed surface integral of the surface's normal vector

Is it true that the surface integral over any closed surface (we are in $R^3$) of the normal vector $\hat n$ of that surface, say $\hat n$ is pointing outward, is zero? In other words, is it true that ...
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1answer
28 views

Show $\left(\vec{A}\cdot\nabla\right)\vec{A} = \nabla\left( \frac{A^2}{2} \right) - \vec{A}\times\left(\nabla\times\vec{A}\right)$

\begin{equation} \begin{aligned} \left(\vec{A}\cdot\nabla\right)\vec{A} &= \nabla\left( \frac{A^2}{2} \right) - \vec{A}\times\left(\nabla\times\vec{A}\right)\\ ...
2
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2answers
68 views

Prove that $\nabla\times (a\vec{A})= a(\nabla \times \vec{A})+(\nabla a)\times A$

$\vec{A}$ is a vector field and each of its component is a function of $x, y$, and $z$: $\vec{A} = u\hat{i} + v\hat{j} + w\hat{k}$ $u = u(x,y,z)$ $v = v(x,y,z)$ $w = w(x,y,z)$ $a$ is a scalar ...
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2answers
13 views

Proof verification of the Triangle Inequality for $k$ vectors?

One of my homework problems was to prove an extension of the Triangle Inequality to $k$ vectors through induction, and I produced a five step proof that I think is correct, but I'm unsure that step ...
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1answer
23 views

Flux of $(0,2y,z)$ over the cylinder (?) $y=\ln(x)$

Let $S$ be the portion of the cylinder $y=\ln(x)$ (what, this is a cylinder?) in the first octant such that the projector parallel to $y$ over the plane $xz$ is the rectangle $1\le x\le e$, $0\le z\le ...
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1answer
41 views

Is this 2-tensor symmetric? It satisfies these conditions

I have some scalar field $p$ and a 2nd order tensor $\textbf{T}$ such that $div( \ \textbf{T} \ \textbf{grad}(p) \ )=0$ $\textbf{curl}(\ \textbf{T} \ \textbf{grad}(p) \ )=\textbf{0} $ $\textbf{T}$ ...
3
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0answers
49 views

Determinant of non-square Jacobian

Suppose I have a 3d solid in ${\bf R}^4$ which can be parametrized by the function $F:W\subset{\bf R}^3\rightarrow{\bf R}^4$. Now suppose I want to calculate the volume of this solid. Then naively I ...
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1answer
22 views

From $||\alpha'||\geqq\alpha'\cdot\mathbf u$, deduce $L(\alpha)\geqq d(\mathbf {p,q})$, where $L(\alpha)$ is the length of $\alpha$

Let $\alpha: [a,b]\to\Bbb R^3$ be an arbitrary curve segment from $\mathbf p=\alpha(a)$ to $\mathbf q=\alpha(b)$. Let $\mathbf {u=\frac{q-p}{||q-p||}}$, the unit vector from $\mathbf p$ to $\mathbf ...
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0answers
17 views

extracting the base of a subspace without any knowledge of it

I would like some help with some basic concepts on linear algebra... Thanks in advance! Vspace = ...
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0answers
15 views

Verify the divergence theorem for $\vec{F}=x\hat{i}-y^2\hat{j}+z^2\hat{k}$ over the region bounded by $x^2+y^2=4,z=0,z=4$

Verify the divergence theorem for the vector function $\vec{F}=x\hat{i}-y^2\hat{j}+z^2\hat{k}$ over the region bounded by $x^2+y^2=4,z=0,z=4$ First, using Divergence Theorem, $$div\vec{F}=(1-2y+2z)$$ ...
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0answers
13 views

evaluate surface integral for vector field (x^2 + y^2 < z^2)

I have the following surface integral problem: Let $S$ be the surface ${S:(x,y,z): x^2+y^2\leq z^2, (0\leq z \leq 2)}$. Evaluate the surface integral for the vector field $A= ...
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1answer
21 views

Find reflection in a cube

Let C be a cube in $R^3$, $C=\{(x,y,z): 0\leq x,y,z,\leq 1\}$. Find a reflection of a diagonal of a face with respect to a plane orthogonal to main diagonal. I am trying to study Vector Calculus by ...
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1answer
26 views

Find the equation of a plane from a line

Let $L$ be a line that passes through points $a = (1,-1,-2)$ and $b =(2,-1,1)$. Let $V_1$ be the plane $x+y-3z+6=0$. Find the equation for $L$. Find the equation for the plane $V_2$ that ...
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0answers
22 views

Verify my calculation of the surface integral without divergence theorem

I have $F=xyi-y^2j+zk$ Over surface $z=0$, $s \le1 $, $x^2+y^2 \le s$ My approach to calculate $ \iint F.ds$ was the outward normal is $k$ the dot product of this with F gives z so integral becomes ...
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2answers
87 views

How to show $\DeclareMathOperator{curl}{curl}\curl\curl(e_r) = 0$

I want to figure out how to calculate $\text{curl}(e_r$). Where $e_r$ is a base vector for the Spherical co-ordinate system. Taking $e_r = (\sin\theta \cos\phi)i+(\sin\theta ...
3
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1answer
27 views

Curl and Product Rule

I know the famous identity $\nabla \times (\vec{A} \times \vec{B}) = A(\nabla \cdot \vec{B}) -B(\nabla \cdot \vec{A}) + (B\cdot \nabla) \vec{A}- (A\cdot \nabla) \vec{B}$ My question: Can I ...
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0answers
45 views

Decomposition into simple bivectors

According to Wikipedia, any element of $\wedge^2\Bbb R^n$ should be decomposable into $n/2$ simple bivectors for $n$ even or $(n-1)/2$ for $n$ odd. How do I count that? How do I check that ...
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1answer
30 views

How to find a potential of a differential form?

I need some help in understanding the meaning of this exercise: Determine a potential of the following differential form $$\omega = (3x^2y + z) dx + (x^3 + 2yz) dy + (y^2 + x) dz$$ I don't ...
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1answer
29 views

divergence of the cross product of two vectors proof

Prove $\vec{\nabla}\left ( \vec{A}\times\vec{B} \right )=\vec{B}\left ( \nabla \times\vec{A} \right )-\vec{A}\left ( \nabla \times\vec{B} \right )$ I have expanded the LHS for this and obtain a ...
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0answers
16 views

Verify Stokes theorem for $F=(x^2-y^2){\bf i}+2xy{\bf j}$ in a rectangular region

Verify Stokes theorem for $F=(x^2-y^2){\bf i}+2xy{\bf j}$ in the rectangular region in the $xy$ plane bounded by the lines $x=0,x=a,y=0$ and $y=b$.
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0answers
13 views

Helmholtz decomposition of $v\in (L^2(\Omega))^3$

Let $\Omega\subset\mathbb{R}^3$ be a bounded domain with Lipschitz boundary $\partial\Omega$ and outward unit normal $n$. I want to study the characterize whether a vector function defined on ...
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0answers
34 views

Understanding calculations for the velocity of a crane

I am not 100% sure this is the best substackexchange to ask my question, but I ll have a try. I found this example where they calculate the velocity of the arm of a crane. But there are some parts ...
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2answers
25 views

Equivalence between two systems of vector equations

I need to solve the system $$\nabla^2 \mathbf{u} = \nabla p \\ \nabla \cdot \mathbf{u} = 0$$ in a subdomain of $\mathbb{R}^3$ with mixed boundary conditions, where $\mathbf{u}$ is vector field and ...
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1answer
33 views

Is this a correct identity for the Kronecker delta and the Alternating Tensor?

If $\varepsilon_{ijk}$ is the alternating tensor and $\delta_{in}$ is the Kronecker delta, am I correct in thinking that $$ \delta_{in}\varepsilon_{ijk} = \varepsilon_{ink} $$ If not, what is the ...
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0answers
18 views

Swapping partial derivative and curl operator

The third Maxwell equation states that $$\nabla \times \mathbf E = -\frac{\partial\mathbf B}{\partial t}.$$ Then I have in my notes: $$\nabla \times (\nabla \times \mathbf E) = -\nabla \times ...
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0answers
12 views

Gaussian theorem: determining a surface and exterior normal from solid figure

Given a vector field: $ \vec{v} = (z^2-x^2-y^2-2) \cdot (x,y,z) $ and a the solid figure $ K = \{(x,y,z) \in \mathbb{R} : z > \sqrt{2+x^2+y^2}, 2 < z < 3 \} $. To determine: $ \int_{K} ...
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1answer
18 views

Why is the expression $u_{j}\frac{\partial u_{j}}{\partial x_{i}}$ equivalent to $\frac{1}{2}\frac{\partial}{\partial x_{i}}(u_{j}u_{j})$?

It says in my lecture notes that the index notation $u_{j}\frac{\partial u_{j}}{\partial x_{i}}$ is equivalent to $\frac{1}{2}\frac{\partial}{\partial x_{i}}(u_{j}u_{j})$, but does not explain why. ...
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1answer
35 views

To evaluate using Gauss Divergence Theorem

Using Gauss Divergence Theorem, evaluate the integral $\int_{S}\int F.\hat n dS$ where $F=(4xz,-y^2,4yz)$ . S is surface of solid bounded by sphere $x^2+y^2+z^2=10$ and paraboloid $x^2+y^2=z-2$ and ...
0
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0answers
18 views

$F=(y^2z^2,z^2x^2,x^2y^2)$ be a vector field , to find a nonzero scalar field $f$ such that $fF$ is a grdient i.e. $curl(fF)=O$?

Let $F=(y^2z^2,z^2x^2,x^2y^2)$ be a vector field . how to find a nonzero scalar field $f$ such that $fF$ is a grdient i.e. $curl(fF)=O$ ? Please help . In general , given a vector field $F$ in ...
0
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1answer
19 views

Application of Gauss Divergence Theorem

Consider $$-\triangle u = f \ \ \ \ \ \text{in} \ \ \ \Omega$$ .$$\frac{\partial u}{\partial n} = g \ \ \ \text{on} \ \ \ \ \partial \Omega $$ Where $\Omega \subset ...
3
votes
2answers
127 views

What is the normal vector to the plane ax+by+cz=d?

If d=0, then we obviously see that the equation is the Euclidean inner product with (a,b,c) and (x,y,z) that equals zero - and so (a,b,c) is the normal vector to the plane. What if $d \ne 0$? Then ...
0
votes
2answers
38 views

Strong maxima and minima

I'm stuck with this problem, in particular at b): Let $u:D \subseteq \mathbb{R}^2 \to \mathbb{R} \in C^2$ a harmonic function. $u$ has a local maximum at point $\vec{p} \in D$. Then: (a) Show that, ...