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).

learn more… | top users | synonyms (1)

3
votes
2answers
335 views

Total Derivative and Multilinear Functions

So I'm starting to work through Spivak's Calculus on Manifolds and I'm having a little trouble verifying some of the claims made in the book problems. To review: Given a function ...
2
votes
0answers
57 views

Difficult multivarate random variable - how to calculate it?

I have a random variable defined by $Y=\frac{\sum_{j=1}^{N}l_j \cos\theta_j}{\sum_{j=1}^{N}l_j\sin\theta_j}$ where $l_j \sim \text{log-normal-distribution} (\left \langle l \right \rangle, \sigma _l)$ ...
2
votes
2answers
81 views

How do I rotate a given point on $\mathbb{S}^n$ so that I send it to the South Pole

This seems pretty trivial but I'm not sure what to do. My coordinates are Cartesian, and I want to send point the $(x_1,...,x_{n-1},x_n)$ to point $(0,...,0,-1)$ so that all the other points are also ...
2
votes
1answer
128 views

Lagrange multiplier constrain critical point

When using Lagrange multipliers in an inequelity, $$ f(x,y) = x^2+y $$ with the constraint $$ x^2+y^2 \leq 1. $$ I have to find the critical points inside the "disk" right? I've done $$ f_x = 2x ...
2
votes
2answers
158 views

Other way to express $e^{|x|+|y|}$

I have a joint PDF with $e^{|x|+|y|}$. I know I can separate the function in two functions, $e^{|x|}$ and $e^{|y|}$. The limits for $x$ and $y$ are from $-\infty$ to $\infty$. Can I integrate from $0$ ...
2
votes
1answer
225 views

Calculate the volume between $z=x^2+y^2$ and $z=2ax+2by$

I'm trying to calculate the volume between the surfaces $z=x^2+y^2$ and $z=2ax+2by$ where $a>0,b>0$. Here's what I've tried: First I noticed the projection of the volume to the xy plane is a ...
2
votes
2answers
762 views

A continuously differentiable map is locally Lipschitz

Let $f:\mathbb R^d \to \mathbb R^m$ be a map of class $C^1$. That is, $f$ is continuous and its derivative exists and is also continuous. Why is $f$ locally Lipschitz? Remark Such $f$ will not be ...
2
votes
1answer
348 views

Find the Jacobian

Find the Jacobian $$\frac{\partial(x,y)}{\partial(u,v)}$$ for $x=u^2+v^2$, $y=u^2-v^2$. My solution: I tried solving it as it is by using the Jacobian matrix (determinant?) and got my answer to be ...
2
votes
2answers
1k views

What is the vector form of Taylor's Theorem?

I checked most of the posts about Taylor expansion with scalar functions. Could anyone tell me what is the multivariate version of Taylor's Theorem, and how I can use it?
1
vote
1answer
63 views

Continuity conditions for multivariate functions.

Is the following true ? A proof or counter-example or reference would be nice. A function $f:\mathbb{R}^2 \rightarrow \mathbb{R}$ is continous at $(0,0)$ if and only if if for all $a, b$, the limits ...
1
vote
1answer
51 views

Solve Multivariate Polynomial

Is there a general way to solve a multivariate polynomial (example here: http://mathworld.wolfram.com/MultivariatePolynomial.html) Say for instance I knew some function $F(x,y) = xy + x^2 + y^3 + ...
1
vote
2answers
92 views

How to make sense of this calculus notation, Advanced College Level

I have $f(x)$=$(2x,e^x)$ what does this notation mean? Notation: $Df(\frac{∂}{∂x})$ Certainly $Df(x)$=$(2,e^x)$ but how can I replace $x$ with $\frac{∂}{∂x}$? Particularly, how can I make sense of ...
1
vote
1answer
132 views

Evaluate the line segment intergal

Evaluate the line integral $$\int_C xe^{y}\, {\rm d}s,$$ where $C$ is the line segment from $(-1,2)$ to $(1,1)$. I do not get this part of calculus at all please show me how this is solved and if ...
1
vote
1answer
246 views

shortest distance between two points on $S^2$

Length of Curve in $2D$ is $l_{\gamma}(\mathbb{R}^2)=\int_{0}^{1}\sqrt{(dr/dt)^2+r^2(d\theta/dt)^2}$ Length of a curve in $3D$ is ...
1
vote
1answer
78 views

Question Concerning Vectors

I am given the information that $ \vec{u} = \langle 1, 1/2 \rangle$ and $\vec{v} = \langle 2,3 \rangle$. There are a few pieces I am asked to find, and these are the one I am having trouble with: ...
1
vote
2answers
194 views

Do we need additional assumptions for problem 3-37 (b) in Spivak´s calculus on Manifolds?

Problem 3-37 (b) reads: Let $A_{n}=[1-1/2^{n},1-1/2^{n+1}]$. Suppose that $f:(0,1)\rightarrow \mathbb{R}$ satisfies $\int_{A_{n}}f=(-1)^{n}/n$ and $f(x)=0$ for any $x\notin$ any $A_{n}$. Show that ...
1
vote
1answer
169 views

Hölder continuity of a function from $[0,1]$ to $[0,1]^2$

I'm trying to prove that if $g: [0,1] \longrightarrow [0,1]^2$ is an $\alpha$-Hölder continuous mapping whose image is the entire square $[0, 1]^2$ then $\alpha \leq 1/2$. I wouldn't know where ...
1
vote
0answers
148 views

Help to show this equality of differential operators in spherical coordinates

There are two points on the same sphere with coordinates ${R, \theta_1, \phi_1}$ and ${R, \theta_2, \phi_2}$. Also I have the operator $\displaystyle { \nabla _{{\Omega _1}}^2 + \nabla _{{\Omega ...
0
votes
2answers
79 views

2 examples to try to understand partials derivatives and deriviability

To prove that a functions has partial derivatives every partial has to exist, and every partial exist only if the limit of definition of partial exist. Is this right? Then if partials exist ,and the ...
0
votes
0answers
19 views

Fréchet normal cone

Given $x\in \Omega(\subset X)$ (X: Banach space) and $\varepsilon\geq 0$, the set of $\varepsilon-$normals to $\Omega$ at $x$ by \begin{align} \widehat N_\varepsilon(x;\Omega):=\left\{x^*\in X^*\mid ...
0
votes
1answer
64 views

Triple Integral Volume Question

The question asks for the triple integral of $e^{-(x^2+y^2+z^2)^{3/2}} dV$ where $D$ is a sphere of radius $4$. The answer that I came up with is $2(1-e^{-64})$. However, I am not confident in this ...
0
votes
0answers
28 views

Lower boundary for hessian eigen values

This question appears on a differential geometry test i am trying to solve. Let $f$ be a function of $x,y$ and $k>0$ a positive real number. the function obeys the following: $f(0,0)=0$ and ...
0
votes
1answer
118 views

Can anyone help me with these double Integrals using mathematica

$$ \int_0^6 \int_0^4 \frac{\sqrt{(1+x^2+y^2 )^2+4 ( x^2+y^2)}}{1+x^2+y^2}\, dy \, dx$$ And $$\int_{-1}^1 \int_{-y}^y \frac{1}{(1+y^2)^2} \sqrt{(1+y^2)^4 + 4x^2(1+y^2)^2 + 4y^2(1+x^2)^2} \, dx \, dy$$ ...
0
votes
1answer
31 views

Multivariate integration of a derivative w.r.t. a single variable

$x=(x_{1},...,x_{n})$. If $\frac{\partial g(x)}{\partial x_{l}}=f(x_{l})$ for $l=1,...,n$, should we have $g(x)=\sum_{l=1}^{n}\int f(x_{l})dx_{l}+c$? If yes, what's the theorem or proposition behind ...
0
votes
1answer
56 views

Multivariable Calculus, Two Path Test.

I'm have trouble understanding how you determine which paths you should choose. In the book for the function lim as (x,y) goes to (0,0) $$f(x,y) = \frac{2x^2y}{x^4+y^2}$$ They say "We examine the ...
0
votes
3answers
405 views

Show discontinuity of $\frac{xy}{x^2+y^2}$

How to show this function's discontinuity? $ f(n) = \left\{ \begin{array}{l l} \frac{xy}{x^2+y^2} & \quad , \quad(x,y)\neq(0,0)\\ 0 & \quad , \quad(x,y)=(0,0) \end{array} ...
0
votes
3answers
79 views

When is $f_{xy}(x,y)\neq f_{yx}(x,y)?$

When is $f_{xy}(x,y)\neq f_{yx}(x,y)?$, where $f_{xy}$ and $f_{yx}$ denote the mixed (second) partial derivatives of a multivariable function $z=f(x,y)$.
0
votes
1answer
86 views

Explanation on a proof of a property of mollifiers

Here are some definitions that was taken of PDE Evans book: Here is a proof of a property of mollifiers: My (elementary) question is: Why is the convergence uniform on $V$? Thanks.
0
votes
3answers
446 views

How does one prove that if function's partial derivative respect to every variable is zero, function is constant?

How does one prove that if function's partial derivative respect to every variable that the function defines over is zero function is constant function? I just noticed it, but I cannot prove it.
0
votes
3answers
204 views

Green theorem of curve

Use Green's Theorem to evaluate the integral $$\int\limits_C \left(y-x\right) \mathrm dx+\left(2x-y\right) \mathrm dy$$ for the path C defined as $x=2\cos\theta \;\text{and}\; y=2\sin\theta.$ Here is ...
0
votes
2answers
86 views

Determining the Moment of inertia

Let $a,b,c$ be positive real numbers such that $c<a$. Suppose given is a thin plate $R$ in the plane bounded by $$\frac{x}{a}+\frac{y}{b}=1, \frac{x}{c}+\frac{y}{b}=1, y=0$$ and such that the ...
0
votes
1answer
271 views

Green's Theorem

Let $C$ be the closed,piecewise curve figured by traveling in straight lines between the points $(-2,1),(-2,-3),(1,-1),(1,5)$ and back to $(-2,1)$, in that order. Use Green's Theorem to evaluate the ...
0
votes
1answer
325 views

Computing $\iint \sin(4x^2 +2y^2)\, \mathrm dA$ over an elliptical region

$$\iint \sin(4x^2 +2y^2)dA,$$ where the region is bounded by ellipse $4x^2 + 2y^2 = \pi$ and the lines $y = 0$ and $y = \sqrt{2}x$. This looks like a change of variables integral. Need some hints. ...
0
votes
1answer
204 views

Multivariable Calculus, Jacobian

In the xy-plane, draw the region R bounded by the lines $y = 1+x, \quad y= -1 +x, \quad y= -1-x, \quad y= 1-x$ Use double integral in rectangular and polar coordinates to find the area of R you ...
0
votes
1answer
141 views

Arc length of level sets

I have a function $z = B \sin x \ \sin y+\cos x \ \cos y$. Where $0 \leq x \leq \pi$ and $0 \leq y \leq \pi$. I need to find the length of the curve that describes a level set for any value of $B$. ...
0
votes
1answer
523 views

Finding a perpendicular line that connects two skew 3D lines at a particular distance?

I have two linear, skew, 3D lines, and I was wondering how I could find a point on each of the lines whereby the distance between the two points are a particular distance apart, and that the vector ...
54
votes
1answer
2k views

What's the largest possible volume of a taco, and how do I make one that big?

Let $f$ be a continuous, even, positive function over some interval $I=[-a,a]$ such that the total arc length of $f$ over $I$ is at least $2$, $f(0)=0$, and $f$ is increasing on $(0,a)$. View the ...
30
votes
2answers
658 views

Evaluating $\int_{0}^{1}\cdots\int_{0}^{1}\left\{\frac{1}{x_{1}\cdots x_{n}}\right\}^{2}\:\mathrm{d}x_{1}\cdots\mathrm{d}x_{n}$

Here is my source of inspiration for this question. I suggest to evaluate the following new one. $$ I_{n}:= \int_{0}^{1} \! \cdots \! \int_{0}^{1} \left\{\frac{1}{x_{1}x_{2} \cdots ...
26
votes
5answers
2k views

What is an intuitive explanation for $\operatorname{div} \operatorname{curl} F = 0$?

I am aware of an intuitive explanation for $\operatorname{curl} \operatorname{grad} F = 0$ (a block placed on a mountainous frictionless surface will slide to lower ground without spinning), and was ...
10
votes
1answer
364 views

Writing Integrals using Differential Forms

Consider some smooth curve $C \subset \mathbb{R^n}$ and $\gamma:[a,b] \subset\mathbb{R}\rightarrow C$ a parametrisation of $C$ and a continuous vector field $K:\mathbb{R^n} \rightarrow \mathbb{R^n}$. ...
8
votes
3answers
3k views

Divergence as transpose of gradient?

In his online lectures on Computational Science, Prof. Gilbert Strang often interprets divergence as the "transpose" of the gradient, for example here (at 32:30), however he does not explain the ...
9
votes
6answers
5k views

Why do we need vectors and who invented it?

It is natural to understand the need for scalars (numbers), but why did we invent vectors? Who invented it and for what? EDIT: As George Lowther pointed out, the problem is too broad; I added the ...
8
votes
2answers
2k views

Geometric intuition behind gradient, divergence and curl

I learned vector analysis and multivariate calculus about two years ago and right now I need to brush it up once again. So while trying to wrap my head around different terms and concepts in vector ...
11
votes
3answers
526 views

What is the motivation for differential forms?

I am that point in my mathematical career where I am learning differential forms. I am reading from M.Spivak's Calculus on Manifolds. So far I have gone over the tensor and wedge products and their ...
11
votes
1answer
487 views

Visualizing Commutator of Two Vector Fields

I'm reading a book on calculus, the part about vector fields on manifolds. It's a nice book, but with a severe drawback --- it has no pictures. I like how vectors are treated algebraically, as ...
8
votes
1answer
615 views

Eigenfunctions of the Laplacian

I am willing to offer a bounty for this one, so I will give you an exact idea of what I need: I am looking for solutions of $$\Delta \Psi(r,\theta)=k^2\Psi(r,\theta)$$ where $k\in \mathbb{R}$. Such ...
6
votes
1answer
701 views

General form of Integration by Parts

This is a question just out of interest to know the power of integration by parts. There are various level of integration by parts. What are some of the most general form of integration by parts? I ...
7
votes
1answer
93 views

dropping a particle into a vector field, part 2

Okay, so earlier I posted this question "dropping a particle into a vector field " as sort of a feeler question as i study line integrals in order to go into surface integrals and eventually ...
6
votes
2answers
328 views

Why does the volume of a hypersphere decrease in higher dimensions? [duplicate]

First let us define an $n$-ball as the euclidean sphere in $\mathbb{R}^n$ including its interior and its surface where $n$ refers to the number of coordinates needed to describe the object (the ...
3
votes
2answers
464 views

Evaluate the Integral using Contour Integration (Theorem of Residues)

$$ J(a,b)=\int_{0}^{\infty }\frac{\sin(b x)}{\sinh(a x)} dx $$ This integral is difficult because contour integrals normally cannot be solved with a sin(x) term in the numerator because of ...