0
votes
0answers
29 views

Surface Integrals, orientation and parametrizations.

I'm trying to solve the following problem: Integrate $f(x,y,z)=(x,y,z)$ over the surface $z=12$ $x^2 + y^2 \leq 25$ I parametrized the surface with $\sigma (r, \theta) = r \sin(\theta), r ...
0
votes
2answers
30 views

vector question assistance

let there be 2 lines: $(2,-3,1) + s(3,-2,1)$ and $(2,-1,-3) +t(3,-2,1)$ which are parallel to each other. find the formula of the plane determined by them. my try: a vector perpendicular to ...
0
votes
1answer
48 views

evaluating surface integral with divergence theory

If I have to calculate the surface integral of $\iint_S A \cdot n\ \mathrm {ds}$ where $A= 3zi-2xj+5x^2zk$ and $S$ is the surface of the cylinder $x^2+y^2=4$ and lying between $z=0$ and $z=4$ in the ...
2
votes
1answer
44 views

Vector calculation question

the points a b c d are concordantly ( 1,2,-3) , (-1,2,1) , ( 0,1,-2) , ( 2,-1,1) find formula of the plane going thorugh d and which is pararlel to plane abc calculate the volume of pyramid abcd. ...
0
votes
0answers
21 views

Vector pyramid question

Suppose we have a pyramid with two vectors known, as well as the angle between them...if we mutpliy their size by each other mutiply by the sine of the said angle and then by sixth, will we get the ...
0
votes
0answers
20 views

Identity proof vector calculus

I encounter massive problems tackling the following math problem. Especially the notation in line #2 with the $\frac{\partial u}{\partial x^2}$ and in line #3 the $ U:]0,R[\times\mathbb R\ $ , ...
-1
votes
0answers
36 views

Surface integral of a cube

can anyone help me out with this question on surface integrals? Evaluate $\displaystyle\iint\limits_{S}A \cdot dS$ where $A= x^2\hat{i}+ y^2\hat{j}+ z^2\hat{k}$ and $S$ is the surface of a cube. $0 ...
2
votes
1answer
71 views

Spivak's “Calculus in Manifolds” problems

I have some troubles with this problems. Problem 1.18: If $A \subset [0,1]$ is the union of open intervals $(a_i,b_i)$ such that every rational number of $(0,1)$ is contained in $(a_i,b_i)$, for ...
1
vote
1answer
60 views

Stoke's Theorem Example - Help?

From Stoke's Theorem: \begin{equation*} \oint_c \textbf{F}\centerdot d\textbf{r}= \int\int_S (\nabla \times\textbf{F})\centerdot d \textbf{S}\end{equation*} Evaluate $\oint _C \textbf{F}\centerdot ...
1
vote
1answer
29 views

Integration of a vector function through a cilinder

I need to integrate $\vec F(x,y,z) = (2x,-3y,z)$ through a surface $\Sigma$. $\Sigma$ is defined as the surface of the cylinder $x^2+y^2=1$ (so without the top and the bottom) that is confined between ...
1
vote
3answers
57 views

Show that no application $f: \mathbb{R}^2 \rightarrow \mathbb{R}$, of $C^k$ class, $k \geq 1$ can be injective

How can I proof this: Show that no application $f: \mathbb{R}^2 \rightarrow \mathbb{R}$, of $C^k$ class, $k \geq 1$ can be injective, i.e., there are $A,B \in \mathbb{R}^2$ such that $A \neq B$ ...
2
votes
1answer
40 views

Line and Surface Integrals

I am stuck on the following question :( $F(x,y,z)=(y+z)i+(x+z)j+(x+z)k$. The sphere $x^2+y^2+z^2=a^2$ intersects the postive x−, y−, and z−axes at points A, B, and C, respectively. The simple closed ...
0
votes
1answer
48 views

Vector Calculus Identity help

I am having some issues with the following question: Prove the following vector calculus identity in $\mathbb{R}^3$, where $f$ is a twice continuously differentiable scalar field and $F$ is a twice ...
1
vote
1answer
47 views

Find the divergence of the following vector fields

Consider an arbitrary vector field $F$ $$\eqalign{F&=F_1\hat{i}+F_2\hat{j}+F_3\hat{k}\\ &=F_{C_1}\hat{e}_\rho+F_{C_2}\hat{e}_{\phi}+F_{C_3}\hat{e}_{z}\\ ...
2
votes
2answers
100 views

Verify Gauss’s Divergence Theorem

I have this assignment which we have not tackled and am getting mixed up in the divergence theorem tutorials like this one ...
0
votes
1answer
58 views

verifying the divergence theorem for the region in $R^3$ bounded by the surfaces $z=1-x^2$, $y=0$, $y=1$ and the x-y plane

I am stuck on the following question. Vector calculus is not a forte of mine. Let V be the region in $R^3$ bounded by the surfaces $z=1-x^2$, $y=0$, $y=1$ and the x-y plane. S is the closed surface ...
1
vote
1answer
44 views

Gauss Divergence Theorem Calculation help

I am having trouble getting my head around what exactly is required in this problem. Let $S$ be an arbitrary piecewise smooth, orientable, closed surface enclosing a region $\mathbb{R}^3$. Calculate ...
0
votes
1answer
53 views

Prove: If $\int_{\phi}\omega = \int_{\psi}\omega$ whenever $\phi$ and $\psi$ have the same endpoints, then $\omega=df$

This is an exercise that I have been assigned for homework. I don't really know how to approach it though. I know that $\int_{\phi}\omega$ only depends on the endpoints $\phi(a)$ and $\phi(b)$ where ...
1
vote
1answer
45 views

Line Integrals in $\mathbb{R}^n$ and Differential 1-Forms

I am really lost in the notation of my book and am looking for some conceptual help. So... for a differential 1-form $\omega$, every value of $x\in \Omega$ maps to a linear function $\omega _x$, ...
0
votes
3answers
32 views

Dot Product and vector length

Hi! I am working on some online homework for my calc2 class that covers the dot product and I am really struggling with this one question. I understood how to solve part a, because we covered that ...
0
votes
2answers
62 views

True or False Stokes'/Gauss Theorem Problems

I'm having difficulties deciding the truths of the two following statements. The first one I believe is false, but I'm not entirely sure and the second one I don't know how to make heads or tales of. ...
0
votes
0answers
12 views

Bubble inside field of charge

There is a charge distribution $\rho(\mathbf x)$ with electrostatic potential $\phi(\mathbf x)$ s.t for $|\mathbf x|<a,$ $\phi=0$ and for $|\mathbf x|=a$, $\phi(\mathbf x)=\Phi$. Show that $\Phi ...
2
votes
0answers
60 views

Green's Theorem

Hey guys I am having difficulties in problem 5. I thought I understood it, but I suppose I was mistaken. I will now explain what I planned to do to solve this problem and where I got stuck. So I ...
0
votes
0answers
26 views

vector field problems

I'm trying to review some problems on vector fields for the final, and would appreciate if someone can tell me whether my answers are right, so I know if I'm doing it correctly: $f$ is a vector ...
0
votes
1answer
31 views

Showing orthogonality of coordinate surfaces are orthogonal for oblate spheroidal co-ordinates.

So oblate spheroidal co-ordinates are defined as: $$x = \cosh R \cosθ \cos φ$$ $$y = \cosh R \cosθ \sin φ$$ $$z = \sinh R \sin θ .$$ To show the coordinate surfaces for $R$, $\theta,\phi$ are ...
0
votes
1answer
28 views

Vectors and Planes

Let there be 2 planes: $x-y+z=2, 2x-y-z=1$ Find the equation of the line of the intersection of the two planes, as well as that of another plane which goes through that line. Attempt to solve: the ...
0
votes
0answers
26 views

Vector Question Help

A plane is determined by $(x,y,z) = (1,-1,0) + t(1,-1,2)$ and point $p(1,2,3)$. find point of intersection of $(1,4,-1)+s(-6,2,-4)$ with this plane. I tried this: given the data plane equation is:$$ ...
1
vote
1answer
108 views

Vector Calculus Surface Integral (Limits of Integration)

I'm currently having trouble with the following problem. I believe that I have most of the problem set up, but I am having trouble finding what the limits of integration should be. $\int\limits_S ...
0
votes
2answers
59 views

Line integral over a curve in the II quadrant

I am lost here: $C = x^2 + y^2 = 4$ from $(0,2)$ to $(-2, 0)$. Calculate $ \ \int_c y^2 ds \ \ $ and give reasons the sign is correct. It's obviously the circular arc going counterclockwise from ...
0
votes
1answer
28 views

Question about vector equations of lines and planes

Find the equation of the line going through the point $(2,-3,4)$ ,and which is perpendicular to the plane $ x+2y + 2z = 13$ So I tried this: the normal of the plane is $(1,2,2)$, random point on the ...
0
votes
2answers
51 views

Vectors Question

I have a question regarding Vectors; Find the equation of the plane perpendicular to the vector $\vec{n}\space=(2,3,6)$ and which goes through the point $ A(1,5,3)$. (A cartesian and parametric ...
1
vote
1answer
54 views

Line integral with Stokes

Let $C$ be curve $(x-1)^2 + (y-2)^2 =4$ and $z=4$ orientated counterclockwise when viewed from high on the z-axis. Let $$\mathbf{F}(x,y,z)=(z^2 +y^2 +\sin ...
1
vote
1answer
39 views

Basic surface integral with Stokes.

Calculate surface integral $\iint_S \nabla \times \mathbf{F} \bullet \mathbf{n} \; dS$ with stokes, when $$\mathbf{F}=\left\langle\frac{5y(z-1)}{6},xz, 6e^{xy}\cos{z}\right\rangle$$ and $S$ is surface ...
0
votes
1answer
29 views

Question with divergence theorem

Calculate flux of field $$\mathbf{F}=(3x^2 y+6)\mathbf{i}+\left(\frac{x y^2 +1}{3}\right)\mathbf{j}+(3yz^2+3)\mathbf{k}$$ in a box where it's opposite angles are in $(1,1,1)$ and $(2,2,2)$ and it's ...
2
votes
1answer
58 views

Question with Stokes theorem

Show with Stokes that $\oint_C (y\mathbf{i}+z\mathbf{j}+x\mathbf{k})\bullet d\mathbf{r}=\sqrt{3}\pi a^2$ when $C$ is intersection of $x^2+y^2+z^2=a^2$ and $x+y+z=0$. My work: $$z=g(x,y)=-x-y$$ ...
1
vote
1answer
150 views

Line integral using Green's theorem.

Let $C$ be parametrization $\mathbf{r}=\sin(t) \mathbf{i}+\sin(2t) \mathbf{j}$, $t \in [0, 2\pi]$. Sketch picture and investigate how the surface orientates. Calculate line integral $\oint_C ...
0
votes
1answer
19 views

With respect tho the $Oxyz$ axes, the temperature of a media is given by: $T=T_0 (1+ax+by)e^{cz}$

With respect tho the $Oxyz$ axes, the temperature of a media is given by: $$T=T_0 (1+ax+by)e^{cz}$$ Where $a, b, c$ and $T (>0)$ are constants. At the origin O, find the direction in which the ...
0
votes
1answer
38 views

Find the derivative of the scalar field: $\Omega=x^2yz+4xz^2$ at the direction of the vector: $(2, -1, -1)$ at the point: $(1, -2, -2)$

Find the derivative of the scalar field: $$\Omega=x^2yz+4xz^2$$ at the direction of the vector: $(2, -1, -1)$ at the point: $P(1, -2, -2)$ Hint: Write the unit vector $\hat n$ at the beginning, qhich ...
1
vote
1answer
41 views

If $\bf a$ is a constant vector field, and $\bf r$ is the position vector, prove that: $\nabla (\bf a\cdot \bf r)=\bf a$

If $\bf a$ is a constant vectorial field (constant magnitude and direction), and $\bf r$ is the position vector, prove that: $$\nabla (\mathbf a \cdot \bf r)=\mathbf a $$
1
vote
1answer
33 views

Vector potential question

If $\mathbf{F}$ and $\mathbf{G}$ are smooth and conservative. Find vector potential $\mathbf{H}$ for $\mathbf{F} \times \mathbf{G}$. I tried to find it like this (kinda brute force-ishly) $$\small ...
1
vote
1answer
53 views

Vector differential identities

Proof that $\nabla\bullet(f(\nabla g\times \nabla h))=\nabla f \bullet(\nabla g \times \nabla h)$. When $f,g$ and $h$ are smooth scalarfields. Can I expand $\nabla\bullet \overbrace{(f(\nabla g\times ...
1
vote
1answer
40 views

Help with surface integral question

Find the surface area of the plane $$x+\frac{1}{\sqrt{2}}y+\frac 14 z=1$$ limited by the coordinate system planes My findings : I suppose we should express the scalar $z=f(x,y) \rightarrow ...
3
votes
0answers
26 views

normal of a function

Function $$\mathbf{r}(u,v)=(2u+2v)\mathbf{i}+(-3+v^2)\mathbf{j}+(2u^2)\mathbf{k}$$ is a parametrization of a surface. What's the normal vector of this surface at the point $(u,v)=(1,-2)$. What I got: ...
0
votes
1answer
40 views

Field lines of vector field

Okey if $\phi(x,y)=\ln(x^2+y^2), (x,y) \neq (0,0)$. Find the field lines for $\mathbf{G}=\nabla \phi$. So $\mathbf{G}=\frac{2x}{x^2+y^2}\mathbf{i}+\frac{2y}{x^2+y^2}\mathbf{j}$ right? To find the ...
0
votes
1answer
32 views

Basic line integral

Let $C$ be curve along surfaces $z=\ln(1+x)$ and $y=x$ from $(0,0,0)$ to $(1,1,\ln(2))$. Calculate the work done by vector field $$\mathbf{F}(x,y,z)=(2x\sin(\pi y)-e^z)\mathbf{i}+(\pi x^2 \cos (\pi ...
1
vote
2answers
27 views

Question on vector fields

Which ones are vector fields? (I checked my answers) Temperature of room at given point The gravitation that object with mass creates (x) The density of an object at given point Function $f: ...
0
votes
1answer
36 views

Fractional change in volume from scale-factor

I was given the following question which I am unable to get a seemingly correct answer from: A body expands linearly by a factor $\alpha$ due to an increase in temperature. Because of the ...
0
votes
1answer
34 views

A particle has the following path: $\vec{r}(t)=t^2\hat{i}+(t^3-4t)\hat{j}$

A particle has the following path: $$\vec{r}(t)=t^2\hat{i}+(t^3-4t)\hat{j}$$At $t_0=2$ the particle fudges (leaves by the tangent). What is the position of the particle at $t=3$? The particle has ...
0
votes
0answers
41 views

Conservative vector field in polar coordinates

Find the potential function of a conservative vector field $\mathbf{F}(r,\phi)=r\sin(2\phi)\mathbf{\hat{r}}+r\cos(2\phi)\mathbf{\hat{\phi}}$ Does my solution seem right: So if ...
1
vote
1answer
29 views

Solve for the tangent plane using the gradient

I am having a hard time finishing this problem up: Consider the surface $4 x^{2} + 9 y^{2} + 4 z^{2} = 17$ and the point $P = \left( 1, 1, 1 \right)$ on this surface. A) Find the outward unit ...