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).

learn more… | top users | synonyms

0
votes
1answer
49 views

Area of square with vertices: $(x_1,y_1),(x_2,y_2),(x_3,y_3),(x_4,y_4)$

If I know that:$$\int_C-ydx+xdy=\boxed{x_1y_2-x_2y_1}$$ So, why the area of square with vertices: $(x_1,y_1),(x_2,y_2),(x_3,y_3),(x_4,y_4)$ is ...
1
vote
3answers
42 views

Showing that $\displaystyle \oint _C xdy-ydx=x_1y_2-x_2y_1$

Let $C$ be the interval from point $(x_1,x_2)$ to point $(x_2,y_2)$ Show that $\displaystyle \oint _C xdy-ydx=x_1y_2-x_2y_1$ My attempt: Acording Green's theoram $\displaystyle \oint _C ...
2
votes
2answers
31 views

Can the calculation of the surface integral of a specific vector field be simplified?

Suppose the two vector fields are $F(x,y,z)=(x^2,0,0)$ and $G(x,y,z)=(0,0,x z)$ respectively. The surface $S$ is a triangle determined by three points $A:(a_1,a_2,a_3)$, $B:(b_1,b_2,b_3)$ and $C: ...
0
votes
1answer
28 views

Finding if $\frac{-yi+xj}{x^2+y^2}$ is a conservative field

Let $\frac{s}{r}$ be a vector field such that: $s=-yi+xj,r=\sqrt{x^2+y^2}$ Is $\frac{s}{r}$ a conservative field? My attempt: $\frac{s}{r}$ is a conservative field $\iff \displaystyle\oint ...
0
votes
1answer
20 views

Finding domain of vector field $\frac{-yi+xj}{\sqrt{x^2+y^2}}$

Let $\frac{s}{r}$ be a vector field such that: $s=-yi+xj,r=\sqrt{x^2+y^2}$ What is the domain $D$ of $\frac{s}{r}$? My attempt: The domain is $\{x,y\mid x^2+y^2>0\}$ Is it correct?
0
votes
1answer
19 views

Find a vector field to calculate the volume of any subset using the flow through its edge.

Find a vector field $v$ on $\mathbb{R}^n$ with wich you can calculate the volume of every open subset with a smooth edge $\Omega\subset \mathbb{R}^n$ using the flow of the vector field through the ...
1
vote
1answer
25 views

Interpretation of Line Integral with respect to discrete variable

In the paper I am reading, (http://arxiv.org/abs/1308.5376), they solve an integral and I am trying to replicate the results. This question is a simplified version of the integral they calculate, I ...
0
votes
0answers
31 views

Calculating the local minimum of a function

Regarding this function $f(x,y)=1007x^2-x^{2014}+(e^y-1+2x^2)^2$. I want to find the strict local minimum of $f$. I started calculating $\nabla$: $\nabla (1007x^2-x^{2014}+(e^y-1+2x^2)^2)$ ...
1
vote
0answers
53 views

Is $\nabla \chi^2 \cdot \nabla^2 (\nabla \chi^2) = 0$ if $\nabla^2 \chi = 0$?

The title says it. Is $\nabla \chi^2 \cdot \nabla^2 (\nabla \chi^2) = 0$ if $\nabla^2 \chi = 0$? $\chi$ is a field in $R^2$. My attempt: I cannot get rid of this term by using any of the vector ...
0
votes
0answers
32 views

How to get the potential and the gradient of this function?

How to calculate the potential $P_3:\mathbb{R}^3\rightarrow\mathbb{R}$ with the $\nabla P_3=f_3$ of the function $f_3:\mathbb{R}^3\rightarrow\mathbb{R}^3$ with $\large ...
0
votes
0answers
36 views

Higher order vector calculus identities

The wikipedia page https://en.wikipedia.org/wiki/Vector_calculus_identities has vector calculus differentiation identities up to third order. Do higher order identities, in particular for fourth order ...
1
vote
1answer
30 views

Point on surface where tangent plane is perpendicular to line.

I'm given the surface $ x^3-2y^2+z^2=27 $ and have to find where the tangent plane is perpendicular to the line described by \begin{align*} x &= 3t-5 \\ y &= 2t+7\\z&=1-t\sqrt2\end{align*} ...
0
votes
0answers
25 views

Independence of Path for Line Integral of Vector Field Perpendicular to Curve

Let $C$ be a simple, piecewise-smooth curve between points $A$ and $B$, as shown below: Let $\vec{f}$ be a vector field that is defined on the curve (shown above in blue). The magnitude of $\vec{f}$ ...
2
votes
1answer
152 views

Integrating over a specific vector field

I am trying to show that the solution of the following integral is as follows: Define the stopping time: $C(a) = \inf(u \ge 0 : H(\pi(0) |\mu)-H(\pi(u) | \mu) > a)$ Where ...
2
votes
2answers
69 views

Divergence of inverse square vector field

After trying to get the divergence of a vector field I got: $${\nabla \cdot \Bigg(\frac{\vec r}{|r|^3}\Bigg)=\frac{\partial}{\partial r}.\frac{1}{r^2}=-\frac{2}{r^3}}$$ But in wikipedia found that ...
0
votes
1answer
58 views

What is the definition of a gradient?

It has been a while since I have done any vector calculus, is this statement true? $\nabla f(x,y,z) = 0 \iff \dfrac{\partial f}{\partial x} + \dfrac{\partial f}{\partial y} + \dfrac{\partial ...
0
votes
0answers
27 views

measurement of acceleration of an object over time

If I have a battery that weighs 26kg and this battery is placed in a car and the car hits a 15 cm bump (Angle 20 degrees) (to slow down the car), is it possible to calculate the speed of the car ...
2
votes
1answer
38 views

How to calculate $\nabla \ln(|\mathbf{u}-\mathbf{v}|)$

I need to calculate:$$\nabla \ln(|\mathbf{u}(\alpha)-\mathbf{v}(\gamma)|)$$where $\mathbf{u}$ and $\mathbf{v}$ are vector valued functions with $\alpha$ and $\gamma$ as independent values and ...
2
votes
1answer
67 views

Vector Calculus Operator $\vec{u} \cdot \nabla$

I just want to double check on this operator and it's properties. It pops up in fluid mechanics often and I just want to be sure about my understanding: 1) $$(\vec u \cdot \nabla)\vec u$$ Is this ...
3
votes
3answers
45 views

Divergence of $\vec{F} = \frac{\hat{\mathrm{r}}}{r^{2}}$

Consider the vector field $$\vec{\mathrm{F}} = \frac{\hat{\mathrm{r}}}{r^{2}},$$ then the divergence of this field is: $$\vec{\nabla}\cdot\left(\frac{\hat{\mathrm{r}}}{r^{2}}\right) = ...
11
votes
3answers
262 views

Arnold Trivium Problem 39

We find in Arnold's Trivium the following problem, numbered 39. (The double integral should have a circle through it, but the command /oiint does not work here.) Calculate the Gauss integral ...
2
votes
1answer
38 views

A classical solution of Poisson's equation is also a weak solution

Let $\Omega\subseteq\mathbb{R}^n$ be a bounded domain and $u\in C^0(\overline{\Omega})\cap C^2(\Omega)$ be a solution of $$\left\{\begin{matrix}-\Delta u&=&f&&\text{in }\Omega\\ ...
2
votes
1answer
42 views

Differential Forms on Surfaces. Show that $N\cdot (\nabla \times V)\eta=d\phi$ on $x(D)$.

Let $M$ be an orientable surface in $\Bbb R^3$ with a unit normal vector field $N$ and let $x: D\to M$ be a patch. Let $\eta$ be a differential 2-form on $x(D)$ defined by $\eta(x_u,x_v)=\pm\|x_u ...
0
votes
1answer
24 views

Conservative force, prove.

I've problem to understand the notation of this problem: "Let x=xi+yj+zk; say if the force F=(x * k)x is conservative and find a potential function". I do not understead how the vector ...
2
votes
2answers
63 views

Proving that every patch in a surface $M$ in $R^3$ is proper.

Problem Prove that if $\mathbf{y}:E\to M$ is a proper patch, then $\mathbf{y}$ carries open sets in $E$ to open sets in $M$. Deduce that if $\mathbf{x}:D \to M$ is an arbitrary patch, then the image ...
3
votes
0answers
154 views

integral of product of curve with itself in three dimensional

I have the next problem: Let $\gamma:[0,1]\to R^3$ differentiable curve piecewise, and let $\Delta_r=\{(s,t)\in [0,1]^2| |\gamma(s)-\gamma(t)|<r\}$, i want to know if: ...
2
votes
0answers
22 views

Simplified Helmholtz decomposition

Suppose we are given a vector field $$\vec{a}= (x+y+z)\vec{i}+y\vec{j}+z\vec{k} \tag{$*$}$$ And we want to write it as a linear combination of a potential field and a solenoidal field, meaning ...
1
vote
2answers
35 views

Vector Identity Question

I am having some trouble with this question regarding vector diffiriential operators. It seems easy and I am not sure what I am missing. The question: Prove: $$ ...
0
votes
1answer
38 views

Computing the Jacobian determinant for a change of variables,

Why do we compute the partial derivatives in terms on the new variables and not with respect to the old variables? Can someone say this intuitively, so that I have the best chance of remembering why? ...
1
vote
1answer
73 views

Formula for the gradient of $F(\rho,\phi,z)$

Suppose $F(\rho,\phi,z)$ is continuously differentiable, I am interested in showing that the maximum directional derivative of $F$ at any point is given by ...
2
votes
0answers
41 views

Stuck while trying to simplifying this PDE identity

This is related to a fluids HW: Let $u(x)$ be incompressible vector field on n-D space, $x\in \Omega\subset R^n$ and let $\theta(x)$ be a scalar field s.t. $\int_{\Omega} \theta(x)dx=0$, $\nabla.u=0$ ...
1
vote
2answers
45 views

Quickest way to determine if a vector field is conservative?

Say I have some vector field given by $$\vec{F} (x,y,z)=(zy+\sin x)\hat \imath+(zx-2y)\hat\jmath+(yx-z)\hat k$$ and I need to verify that $\vec F$ is a conservative vector field. What would be the ...
3
votes
0answers
24 views

Any unit speed reparametrization $\beta=\alpha(h)$ of $\alpha$ is reparametrized by an arc length function.

These are definitions given from Barret O'neill's Elementary Differential Geometry. Definition $1$. A curve in $\mathbb{R}^3$ is a differentiable function $\alpha: I\to \mathbb{R}^3$ from an open ...
0
votes
0answers
25 views

Prove the following identitie

Given the vector fields F and G in $R^3$, I have learnt the grad(vector of derivatives) or del and curl(cross product) of a function. But I get stack when trying to prove the following identities ...
0
votes
1answer
22 views

What is nabla scalar (a.u) where a is a scalar field and u a vector field?

We have a domain D of say R² and a function a from D to R and a function u from R² to R² what is Nabla dot (au) ? If u were from R² to R we could have simply used the product rule
0
votes
0answers
26 views

The proof by partial derivatives and vector calculus

Prove that if $f(x,y,z)$ is a composite function $F(u)$, where $u=g(x,y,z)$. I am trying to show that $\nabla f=F'(u)\nabla g$. I have learnt the vector calculus and vector fields but the composite ...
1
vote
1answer
24 views

vector triple product does not go from $[\dot{M} \times [H \times M]]+[M \times [H \times \dot{M}]]$ to $2\dot{M}(H,M)$

I have the following term $ [M \times [H \times M]] $ under the time derivative. After using the rule of the derivate of the product I get: $$[\dot{M} \times [H \times M]]+[M \times [\dot{H} \times ...
1
vote
2answers
24 views

Computing partial derivatives using three implicitly defined equations

The three equations $x^2-y\operatorname{cos}(uv)+z^2=0$ $x^2+y^2-\operatorname{sin}(uv)+2z^2=0$ $xy-\operatorname{sin}u\operatorname{cosv}+z=0$ define $x,y,z$ as functions of $u,v$. Compute the ...
1
vote
2answers
62 views

Find a unit tangent vector to a curve that is an intersection of two surfaces.

The intersection of the two surfaces given by the Cartesian equations $2x^2+3y^2-z^2=25$ and $x^2+y^2=z^2$ contains a curve $C$ passing through the point $P=(\sqrt{7},3,4)$. These equations may be ...
0
votes
1answer
30 views

Showing that $\nabla (\alpha f) = \alpha \nabla f$ for constant $\alpha$

I want show that del of alpha times a vector function for is equal to alpha times del of fun using. Alphar is a constant hence it should be factories out after finding partial derivetives,but how do ...
0
votes
1answer
29 views

How to determine the maximum rate of increase in temperature

Suppose that the temperature at a point $(x,y,z)$ in space is given by $T(x,y,z)=\frac{80}{1+x^2+2y^2+3z^2}$ where $T$ is measured in degrees celsius and $x$,$y$ and $z$ in meters. In which ...
1
vote
1answer
21 views

Explicitly demonstrating Stokes' theorem over a tetrahedron.

Consider the vector field: $$\vec{F} = \left(2x-y,-yz^2,-y^2z\right)$$ We are to explicitly show stokes theorem: $$\oint_\Gamma \vec{F}\cdot d\vec{x} = \int\int_{\partial V}\nabla \times \vec{F}\cdot ...
1
vote
1answer
22 views

Showing that a function is the gradient of another function

How do I show that this function; $ f = \frac{\vec{r}-\vec{X}t}{|\vec{r}-\vec{X}t|^3}$ $\vec{X} = (x_1,x_2,x_3)$ and $\vec{r} = (x,y,z)$ is the gradient of another function? like so: $ f = \nabla ...
4
votes
2answers
19 views

Problems on orthogonality and tangency in 3-space.

Find the set of all points $(a,b,c)$ in 3-space for which the two spheres $(x-a)^2+(y-b)^2+(z-c)^2=1$ and $x^2+y^2+z^2=1$ intersect orthogonally. A cylinder whose equation is $y=f(x)$ is tangent to ...
2
votes
1answer
21 views

Let $\mathbf{r}=(x,y,z)$,$r=||\mathbf{r}||$. Show the following equation on $B\cdot \nabla (A\cdot \nabla (\frac{1}{r}))$

Let $\mathbf{r}=(x,y,z)$ and let $r=||\mathbf{r}||$. If $A$ and $B$ are constant vectors show that: $$B\cdot \left(\nabla \left (A\cdot \nabla \left(\frac{1}{r}\right)\right)\right)=\frac{3A\cdot ...
1
vote
3answers
45 views

If $f(x,y,z)=(r \times A)\cdot (r\times B)$, where $r=(x,y,z)$, show that $\nabla f(x,y,z)=B\times (r\times A)+A\times (r\times B)$.

If $f(x,y,z)=(r \times A)\cdot (r\times B)$, where $r=(x,y,z)$ and $A$ and $B$ are constant vectors, show that $\nabla f(x,y,z)=B\times (r\times A)+A\times (r\times B)$. I'm a bit lost on this ...
0
votes
1answer
14 views

If $||\nabla f(x,y)||^2=2$, determine constants $a$ and $b$ such that $a(\frac{\partial g}{\partial u})^2-b(\frac{\partial g}{\partial v})^2=u^2+v^2.$

The change of variables $x=uv$, $y=\frac{1}{2}\left(u^2-v^2\right)$ transforms $f(x,y)$ to $g(u,v).$ If $\left\|\nabla f(x,y)\right\|^2=2$ for all $x$ and $y$, determine constants $a$ and $b$ such ...
2
votes
1answer
28 views

Concerning an application of the divergence theorem

I was studying the derivation of Helmholtz decomposition through Wikipedia and I've come across an application of the divergence theorem which I'm not familiar with. I'd appreaciate if you could help ...
2
votes
1answer
25 views

show that the equation $r_1+r_2= \text{constant}$ implies the relation $\mathbf{T}\cdot \nabla(r_1+r_2)=0,$

This is a problem from Apostol's Calculus, which I have difficulty solving. If $r_1$ and $r_2$ denote the distances from a point $(x,y)$ on an ellipse to its foci, show that the equation $r_1+r_2= ...
0
votes
1answer
22 views

Find a vector $V(x,y,z)$ normal to the surface $z=\sqrt{x^2+y^2}+(x^2+y^2)^{3/2}$

(a) Find a vector $V(x,y,z)$ normal to the surface $$z=\sqrt{x^2+y^2}+(x^2+y^2)^{3/2}$$ at a general point $(x,y,z)$ of the surface, $(x,y,z)\neq (0,0,0)$. (b) Find the cosine of the angle $\theta$ ...