Use this tag for questions about differential and integral calculus with more than one independent variable. Some related tags are (differential-geometry), (real-analysis), and (differential-equations).

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1answer
12 views

Prove that the image of a curve has zero content

Definition: A set $A \subset \mathbb{R}^2$ is said to have zero content if, for all given $\varepsilon >0$, exists a finite collection of rectangles $A_1, \dots, A_n$ such that $A \subset ...
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0answers
11 views

Expanding a multivariable function [on hold]

How can I expand a function $$ f(x,y) = \dfrac{x+y}{x^2 + y^2} \ln(1+xy) $$ to have a total differential everywhere on $\mathbb{R}^2$?
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0answers
11 views

What is the double integral of sigmoidal function from minus infinity to plus infinity? [on hold]

The integral of sigmoidal function from $0$ to $\infty$ is available but I need integral from $-\infty$ to $\infty$.
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1answer
17 views

Suppose $S_1 =\{ u_1 , u_2 \}$ and $S_2 = \{ v_1 , v_2 \}$ are each independent sets of vectors in an n-dimensional vector space V..

Let us assume that every vector in S_2 is a linear combination of vectors in S_1. Question: Does that mean that S_1 and S_2 are bases for the same subspace of V? I know that the answer to this ...
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2answers
40 views

Suppose $S_1 =\{ u_1 , u_2 \}$ and $S_2 = \{ v_1 , v_2 \}$ are each independent sets of vectors in an n-dimensional vector space V.

Let us assume that every vector in $S_2$ is a linear combination of vectors in $S_1$. Question: Does that mean that $S_1$ and $S_2$ are bases for the same subspace of $V$? I know that the answer to ...
3
votes
3answers
41 views

Curl of a vector field cross itself?

Is there a neat expression for $(\nabla \times f ) \times f$ for some vector field f? Here is my attempt at a solution: $((\nabla \times f ) \times f)_i = \epsilon_{ijk}(\nabla \times f )_jf_k$ $ = ...
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3answers
55 views

Determine whether the set $\{v_1 + v_2 - v_3, 2v_1 + 2v_3, -v_1 + v_2 - 3v_3\}$ is linearly dependent or independent.

We had a question on our last test that was very similar to this and I only got $2$ points of $6$ and I want to make sure I do it right this time. Here's my solution to that one: Let $v_1, v_2,$ and ...
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4answers
28 views

Let V be a vector space and W a subset of V. Suppose zero is in W and W is closed under addition. Is W a subspace of V?

I know that the answer to this question is No. My question is why is the answer no? What's missing? if possible give a specific example of both V and W such that W satisfies above conditoins but it ...
0
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1answer
20 views

Existence of Solutions to PDEs - How do I know I've got them all?

I'm taking a very computational course in partial differential equations. Because of this emphasis, I'm feeling very underwhelmed by the course, and have a lot of questions that really aren't ...
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0answers
10 views

Limits of this parametrisation

Use Stokes’ theorem to solve the following integral (each time the curve is oriented counterclockwise when viewed from above). $$\int \limits_C (x+2y)dx+(2z+2x)dy+(z+y)dz$$ where $C$ is the ...
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0answers
62 views

Random walk, expectation & variance, joint probability, approximation question [on hold]

Consider the following random walk on a plane: The walk commences at the origin and at each timestep, a step of unit length is taken in a random direction $\theta$ (measured relative to the positive x ...
1
vote
1answer
29 views

Spherical coordinates for sphere with centre $\neq 0$

For something like $(x-1)^2+y^2+z^2=1$, we would let $x-1=\rho \sin (\phi)\cos (\theta)$ $y=\rho \sin (\phi)\sin (\theta)$ $z=\rho \cos (\phi)$ But we know $\rho =1$ right? So it becomes $x-1= ...
3
votes
0answers
21 views

Focal points of the parabola $y = x^2$ in $\mathbb{R}^2$. [on hold]

Let $X$ be an $n-1$ dimensional submanifold of $\mathbb{R}^n$, a "hypersurface." A point in $\mathbb{R}^n$ is called a focal point of $X$ if it is a critical value of the normal bundle map $h: N(X) ...
4
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0answers
20 views

Immersion except at the origin. [on hold]

Whitney showed that for maps of two-manifolds into $\mathbb{R}^3$, a typical cross cap looks like the map $(x, y) \to (x, xy, y^2)$. Prove that this is an immersion except at the origin.
1
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1answer
26 views

write $\iiint_E \hspace{1mm}dV$ in 6 forms. where $E = \left\{ (x, y, z)|0\leq z\leq x+y, x^2\leq y\leq \sqrt{x},0\leq x\leq 1\right\}$

write $\iiint_E \hspace{1mm}dV$ in 6 forms. where $E = \left\{ (x, y, z)\hspace{1mm}|0\leq z\leq x+y, x^2\leq y\leq \sqrt{x},0\leq x\leq 1\right\}$ As you can see two forms are easy. $$\iiint_E ...
-1
votes
1answer
11 views

questions related to connected set, smooth simple curve in $\mathbb{R}^2$ [on hold]

I have following questions that I struggle with. Prove that the boundary of any connected subset of $\mathbb{R}^2$ is a simple curve, or give a counter example. Prove that if $C$ is a smooth simple ...
0
votes
1answer
13 views

Evaluate double integral $\int\int F*d$s where $f(x,y,z)=<2x,2y,z>$ and S is the first octant part of $x+y+z=1$ oriented upward

Evaluate $\int\int Fds$ where $f(x,y,z)=<2x,2y,z>$ and S is the first octant part of $x+y+z=1$ oriented upward How do you solve this? I think you have to parametrize it but I'm not even sure ...
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0answers
13 views

Find the volume of the solid in the first octant bounded by the coordinate planes $y=x^2$ and $y+z=1$

Find the volume of the solid in the first octant bounded by the coordinate planes $y=x^2$ and $y+z=1$ I know its a double integral but i want to make sure i have the bounds correct. $0<x<1$ and ...
0
votes
1answer
32 views

Find local max min or saddle points for $f(x,y)=3y-y^3-3(x^2)y$

Find local max min or saddle points for $f(x,y)=3y-y^3-3(x^2)y$ I know how to solve this problem, just having trouble finding critical pts when i set $f_x = 0$ and $f_y =0$. We have: $$f_x=-6xy = 0$$ ...
0
votes
1answer
22 views

directional derivative problem

for a point M(4,1) and a function $z = x y^2 - (x^2/y^3)$ I was tasked with finding a directional derivative in the direction which creates a 30 degree angle with the $x$ axis....I find it a little ...
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2answers
20 views

Jacobi's Derivative of the Determinant

I've been given the following theorem for the derivative of the determinant of a matrix: "Let $A\in \mathbb{R}^{n\times n}$ be a square matrix. Then the Fréchet derivative of det$: ...
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0answers
11 views

How can I do this line integral using stoke's theorem??

$$ \int_{C}{(y^2+z^2)dx}+(x^2+z^2)dy+(x^2+y^2)dz $$ where C is the intersection of hemisphere $x^2 + y^2 + z^2 = 2ax, z \geq 0$ and $x^2 + y^2=2bx $ where 0 < b < a. Compute line integral ...
1
vote
1answer
15 views

Green's theorem in divergence form and its line integral?

$$ \int_C F \times da $$ $$ k\iint_R \operatorname{div} F \ dx \, dy $$ Hi Let $F$ be two-dimensional vector field. State a definition for the vector-valued line integral so that your definition ...
1
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1answer
17 views

If I take partial derivatives of function using chain rule, how do I remove variable from the partials in the equation?

I have the function $$f(tx,ty)$$ and I want to take the partial derivative of this with respect to $t$. So set $x'=xt$ and $y'=yt$. I applied chain rule and got $$\frac{\partial f}{\partial t} = ...
0
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1answer
24 views

Did Spivak leave out Jordan-measurability too in his definition of partition of unity?

This is a continuation of these two questions that are asking the same thing as each other: An application of partitions of unity: integrating over open sets. Is this definition missing some ...
2
votes
2answers
26 views

Find $\nabla \cdot (f\textbf r)$ and $\nabla \times (f\textbf r)$ of the function $f(x,y,z) = (x^2+y^2)\log(1-z)$

I have been given the function $f(x,y,z) = (x^2+y^2)\log(1-z)$ and I need to find the divergence $\nabla \cdot (f\textbf r)$ and curl $\nabla \times (f\textbf r)$ where $\textbf r$ is the position ...
1
vote
1answer
23 views

Fundamental theorem of calculus in multivariable calculus

I'm not sure if this is the right name for it but with the theorem: Let $f:\sigma \rightarrow \mathbb{R} $ be a smooth scalar field and assume $r: [a,b] \rightarrow \mathbb {R}^n$ is a piecewise ...
1
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1answer
23 views

Calculating line integrals

The curve $\gamma$ is parameterized by: $t \rightarrow(\cos t, \sin t), t \in [0,2\pi]$ I want to calculate the following integrals and I am supposed to explain what "type" of integral each one is. ...
2
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2answers
37 views

Need help solving complicated integral $\oint_{\mathcal C}\begin{pmatrix}x_2^2 \cos x_1 \\ 2x_2(1+\sin x_1)\end{pmatrix} dx$

Let $\mathcal C$ be the curve that traces the unit circle once (counterclockwise) in $\mathbb R^2$. The starting- and endpoint is (1,0). I need to figure out a parameterization for $\mathcal C$ and ...
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0answers
20 views

Integrating 2 form over torus

Let $\Bbb M^2 ⊂ \Bbb R^3$ be the torus of revolution obtained by rotating the circle $(x−2)^2 +z^2 = 1$ in the $xz$ plane around the $z$ axis. Consider the orientation on $M$ induced by the ...
2
votes
2answers
25 views

Evaluating the triple integral $\iiint \limits_R ze^{-(x^2+y^2+z^2)} \, \, dV$

Evaluate the following triple integrals as a repeated integral using an appropriate coordinate systems: $$\iiint\limits_R ze^{-(x^2+y^2+z^2)} \, \, dV ,$$ where $$R=\{ (x,y,z): \, x,y \in (-\infty, ...
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votes
0answers
10 views

Limit of a matrix valued function

The limit of a vector valued function $f:\mathbb{R} \to \mathbb{R^n}$ is defined as: $$\lim_{x \to a} f(x)=(\lim_{x \to a}f_1(x),\dots,\lim_{x \to a}f_n(x))$$ provided that the limits of $f_i(x)$ ...
4
votes
0answers
48 views

Convex set of derivatives implies mean value theorem

Let U$ \subset$ $R^{^{n}}\ $be open, $f:U\rightarrow R^{m}$ differentiable on U, and segment $[a,b]\subset U$. Assume that the set of derivatives $\{ f'(x)\in L(R^{^{n}},R^{^{m}}):x\in [a,b] \}$ ...
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0answers
39 views

divergence and curl of the function $(x^2+y^2)\log(1-z)$

I have been given the function $F(x,y,z) = (x^2+y^2)\log(1-z)$ and I need to find the divergence and curl. I understand that $$\nabla \cdot F = \frac{\partial F_x}{\partial x} + \frac{\partial ...
0
votes
0answers
14 views

Orthogonal surfaces

Prove that the three surfaces of the family $xy/z=u$ $\sqrt{x^2+y^2}+\sqrt{y^2+z^2}=v$, $\sqrt{x^2+y^2}-\sqrt{y^2+z^2}=w$ that pass through just one point are orthogonal I´m assuming that first I ...
0
votes
2answers
52 views

Differentiability and continuity at the origin of piecewise defined $g(x,y) = y-x^2$, $y+x^2$, or $0$

$$g(x,y)= \begin{cases} y-x^2, & y\ge x^2\\ y+x^2, & y\le -x^2\\ 0 & \text-x^2\le y\le x^2 \end{cases}$$ I need to find all the directional derivatives at the origin in the tangent ...
1
vote
1answer
20 views

Smooth manifold

$A=M\cap N$, $M={(x,y,z\in\Bbb R^3)| x^2+y^2=1}$, $N=(x,y,z)\in \Bbb R^3|x^2-xy+y^2-z=1$. $1$. Is $A$ is smooth manifold? $2$. Find the points of $A$ that are farthest from the origin. This is ...
4
votes
2answers
48 views

How to calculate this multivariable limit?

$$ \lim_{(x,y,z)\to (0,0,0) } \frac{\sin(x^2+y^2+z^2) + \tan(x+y+z) }{|x|+|y|+|z|} $$ I know the entire limit should not exist. In addition, the limit: $$ \lim_{(x,y,z)\to (0,0,0) } \frac{\tan(x+y+z) ...
1
vote
1answer
26 views

Center of mass and moment of inertia of a $2$-dimensional donut?

I have a an assignment and I'm stuck on this question: First of all I can't figure out the equation for a $2$-dimensional donut as shown in the diagram. For the calculation of the center of mass, I ...
0
votes
4answers
47 views

Remember the implicit function theorem

First, I know the implicit function theorem, but unfortunately I always have to look it up again and again. If $F(x,y)=0$ then I always forget whether I have to invert the first matrix of the Jacobian ...
0
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0answers
20 views

Find global minimum of the function

I need to find the global minimum of the function $$f ( x) = \langle Ax,x \rangle + 2\langle b ,x\rangle+c$$ where $c \in \mathbb{R}$ is constant, $b \in \mathbb {R}^n$, and $A$ is a positive ...
0
votes
1answer
24 views

Calculate the level set of $(x^2+y^2)\log(1-z)$

I need help with the following question if possible. I'm not entirely sure on how I should begin. Find the level set of $f$ which has value (“height”) $w_0 = 0$, and describe it in words and set ...
0
votes
1answer
17 views

Rate of Change of a Multivariable Function

The problem says, Find the rate of change of $$(x,y,z) = x/z + y/z$$ with respect to t along the curve $$r(t) = \sin^2{t}[ i] + \cos^2{t}[j] + 1/(2t)[k]$$ The answer is apparently ...
0
votes
1answer
18 views

Calculate the taylor polynomial $f(x,y,z) = (x^2+y^2)log(1-z)$

I have been given the following question that I need help with. I have calculated grad f but I'm just not sure exactly how to calculate the taylor polynomial which is part c. Consider the function ...
2
votes
0answers
55 views

Another proof of Inverse Function theorem in $\mathbb{R}$

(Inverse Function theorem in $\mathbb{R}$) Suppose $I\subset \mathbb{R}$ is an open interval and $f:I\rightarrow\mathbb{R}$ is a differentiable function.If for all $x\in I$ is such that $f^{'}(x)\ne ...
2
votes
1answer
34 views

Area under the curve described by θ=ar

I'm interested in finding the area under the curve described by θ=ar, which is a linear curve with slope 'a' in polar coordinates. Here is what the curve looks like: ...
0
votes
1answer
39 views

Explicitly find all pairs $(a,b)$ such that $a^{1/a}=b^{1/b}$ and $a\ne b$.

Explicitly find all pairs $(a,b)$ s.t. $a^{1/a}=b^{1/b}$ and $a\ne b$. My multivariable calculus teacher posed this question as a fun brain teaser for the end of the semester. He said it was ...
1
vote
1answer
35 views

Double integral of $\arctan(x + y)$?

I would like to find $\int_a^b\int_a^b\arctan(x+y)dydx$ I can "simplify" the integration down to $\int_a^b ((x+b)\arctan(x+b)-\frac{1}{2}\ln(1+(x+b)^2) - ...
0
votes
0answers
12 views

Computing the index around a curve with respect to a field, invariance?

If I understood the course book Nonlinear Dynamics and Chaos right, The index can be found by $$\newcommand{\dd}{\mathrm{d}} \newcommand{\id}{\mathrm{d\,}} I_{C}=\frac{1}{2\pi}\oint_C ...
1
vote
1answer
28 views

Convergence rate - Convex optimization

What is the best known algorithm in terms of convergence rate for unconstrained convex optimization and under what assumptions? $\min_{x} f(x)$ where $f(x)$ is a given twice differentiable convex ...