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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27 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 $\nabla$: $\nabla (1007x^2-x^{2014}+(e^y-1+2x^2)^2)$ ...
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46 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 ...
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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 ...
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34 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 ...
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1answer
14 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*} ...
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21 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 $\vec{f}$ ...
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1answer
74 views
+50

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 ...
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2answers
45 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 ...
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1answer
52 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 ...
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25 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 ...
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1answer
36 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
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1answer
58 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
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3answers
41 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) = ...
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3answers
250 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 ...
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1answer
31 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\\ ...
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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 ...
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1answer
23 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 ...
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2answers
54 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
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0answers
150 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: ...
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0answers
20 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 ...
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2answers
33 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: $$ ...
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0answers
17 views

Some verification or correction on vector functions

If phi=XY+YZ+ZX and A= , find at (3,-1,2) (i) A.del(phi) (ii) phi(del).A (iii) (del(phi))xA My calculations shows (i) 49 (ii) 34 (iii) 18i+15j+16k But the book is providing (i) 25 (ii) 2 (iii) ...
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1answer
34 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? ...
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1answer
58 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
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0answers
39 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. $\int_{\Omega} \theta(x)dx=0$, $\nabla.u=0$ ...
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2answers
40 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
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0answers
20 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 ...
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24 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 ...
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1answer
15 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
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22 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 ...
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1answer
19 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 ...
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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 ...
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2answers
31 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 ...
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1answer
29 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 ...
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1answer
26 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 ...
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1answer
20 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 ...
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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 ...
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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 ...
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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 ...
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3answers
42 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 ...
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1answer
13 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
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1answer
27 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
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1answer
24 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= ...
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15 views

Find a pair of linear Cartesian equations for the line which is tangent to both the surfaces $x^2+y^2+2z^2=4$ and $z=e^{x-y}$ at the point $(1,1,1)$.

Find a pair of linear Cartesian equations for the line which is tangent to both the surfaces $x^2+y^2+2z^2=4$ and $z=e^{x-y}$ at the point $(1,1,1)$. I think the line that is tangent to both the ...
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1answer
19 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 $\theta$ ...
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3answers
31 views

Evaluate the directional derivative of $f$ for the points and directions specified

Evaluate the directional derivative of $f$ for the points and directions specified: (a) $f(x,y,z)=3x-5y+2z$ at $(2,2,1)$ in the direction of the outward normal to the sphere $x^2+y^2+z^2=9.$ (b) ...
2
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3answers
36 views

Potential for integration

I have the following function inside an integral $$\frac{2xdx + 2ydy + 2zdz}{x^2 + y^2 + z^2}$$ I need to find the potential for solving the integral, but I don't know how to transform it into a ...
2
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0answers
46 views

Area of a GREEN-region

a) Show that the area of GREEN-region B (which is defined by: $A\subseteq R^2$, $A=B_1\cup ...\cup B_m$, and $\mathring{B}_i\cap \mathring{B}_j=\emptyset$) in the plane is defined by: ...
3
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1answer
69 views

Find the flux across a part of the surface $\frac{x^2}{4}+\frac{y^2}{9}+\frac{z^2}{16}=1$

Consider the vector field $$F(x, y, z)= \frac{(x{\rm i} + y{\rm j} + z{\rm k})} {(x ^2+ y ^2 + z ^2)^\frac{3}{2}},$$ and let $S$ be the part of the surface ...
0
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1answer
31 views

The area of surface obtained by rotating a rectifiable curve

Let $\Gamma :X=\gamma(t),a\leq t\leq b$ be a rectifiable parameterized curve in the $(x,z)$-plane of $R^3$, which means $\gamma:[a,b]\to R^3$ is a $C^1$-mapping with $\gamma(t)=(x(t),0,z(t))^T$ and ...