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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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 ...
9
votes
1answer
855 views

Nice way of thinking about the Laplace operator… but what's the proof?

Here's a nice fact: roughly speaking, the Laplace operator gives you the difference between the value of a function at a point and the average value at "neighboring" points. More precisely, in ...
8
votes
2answers
1k 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 ...
5
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 ...
10
votes
1answer
415 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
6answers
4k 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 ...
5
votes
1answer
509 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 ...
10
votes
3answers
353 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 ...
8
votes
1answer
546 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
2answers
263 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
457 views

Prove or disprove this calculus limit result by geometric approach

my question is: Could we prove the this conversion of variable work by my formula on the bottom? $$\iint_R f(r,\theta) \ dxdy = \int_a^b \int_0^{r(\theta)} f(r,\theta) r (dr)\ d\theta$$ as $d ...
2
votes
4answers
822 views

Calculating the max and min of $\sin(x)+\sin(y)+\sin(z)$

I took the partial derivatives of $\sin(x)+\sin(y)+\sin(z)$ and it didn't work out, so I am trying to use Lagrange's method (with the constraint: $x+y+z=\pi$)... I am not sure how to set this up. ...
9
votes
1answer
141 views

How to calculate this volume?

Be the sets: $$C:= \lbrace (x,y,0)\in\mathbb{R}^{3}: (x-1)^2+y^2=1\rbrace$$ $$C':= \lbrace (x,0,z)\in\mathbb{R}^{3}: (x+1)^2+z^2=1\rbrace $$ $$\overline{C}= \lbrace tx+(1-t)x': x\in C, x' \in C', t\in ...
7
votes
3answers
601 views

Surface integral over ellipsoid

I've problem with this surface integral: $$ \iint\limits_S {\sqrt{ \left(\frac{x^2}{a^4}+\frac{y^2}{b^4}+\frac{z^2}{c^4}\right)}}{dS} $$, where $$ S = \{(x,y,z)\in\mathbb{R}^3: \frac{x^2}{a^2} + ...
7
votes
3answers
574 views

Proof for the funky trace derivative : $d (\operatorname{trace} (ABA'C))$?

Given three matrices $A$, $B$ and $C$ such that $ABA^T C$ is a square matrix, the derivative of the trace with respect to $A$ is: $$ \nabla_A \operatorname{trace}( ABA^{T}C ) = CAB + C^T AB^T $$ ...
6
votes
1answer
80 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 ...
5
votes
0answers
186 views

How to prove following integral equality?

Let's have the equality $$ \int \limits_{-\infty}^{\infty} \left[ [\nabla_{\mathbf r'} \times \mathbf A (\mathbf r' )] \times \frac{\mathbf r' - \mathbf r}{|\mathbf r' - \mathbf r ...
5
votes
3answers
1k views

Could you a give a intutive interpretation of curl?

Could you a give a intuitive interpretation of curl, geometrical interpretation, real-world example or physical interpretation would be ok. EDIT: Consider a specific vector field $$\mathbf{v} = ...
4
votes
5answers
3k views

Finding shortest distance between a point and a surface

Consider the surface $S$ (in $\mathbb R^3$) given by the equation $z=f(x,y)=\frac32(x^2+y^2)$. How can I find the shortest distance from a point $p=(a,b,c)$ on $S$ to the point $(0,0,1)$. This is ...
3
votes
2answers
452 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 ...
3
votes
1answer
198 views

Singular manifold of the Jacobian

Suppose I have a map $f: \mathbb R^{N} \mapsto \mathbb R^{N}$ of multivariate polynomial form of degree $K$: $$ f^i: X \mapsto A^{i}_0 + A^{ij}_1 X^{j} + A^{ijk}_2 X^j X^k + \ldots + A^{i i_1 \cdots ...
7
votes
1answer
2k views

Multivariable calculus: hard problems with solutions

I'm practicing for my multivariable calculus exam and I'm having some trouble mostly because I have no way of knowing if my solutions are correct or not. For example, a typical problem goes like ...
6
votes
1answer
208 views

Show $\nabla\cdot\left(\mathbf{F}\times\mathbf{G}\right)=\mathbf{G}\cdot(\nabla\times\mathbf{F})-\mathbf{F}\cdot(\nabla\times\mathbf{G})$

Question as follows. Suppose that $\mathbf{F}$,$\mathbf{G}:\mathbb{R^3}\rightarrow\mathbb{R^3}$ and $\phi:\mathbb{R^3}\rightarrow\mathbb{R}$ are smooth. Show using the summation convention that ...
5
votes
1answer
130 views

Extended matrix function

I have a continuous matrix-valued function $f:\mathbb{R}^d\mapsto {\cal M}_{k\times d}$, with $d<k$, such that $f(x)$ is full rank for all $x\in\mathbb{R}^k$. Can I extend this function to be a ...
5
votes
1answer
84 views

Decomposite a vector field into two parts

Let A be a region in $\mathbb R^3$, and suppose $ \vec {\mathbf F}$ is a smooth vector field on A. I was asked to show that I can write $\vec {\mathbf F}=\vec {\mathbf F_1}+\vec {\mathbf F_2}$, s.t. ...
5
votes
3answers
473 views

Proving a function is continuous at $(0,0)$.

How would you prove or disprove that the function given by $$f(x,y) = \begin{cases} \frac{x^2y}{x^2 + y^2} & (x,y) \neq (0,0) \\ 0 & (x,y) = (0,0) \end{cases}$$ is continuous at $(0,0$)?
4
votes
2answers
470 views

Why isn't there a continuously differentiable injection into a lower dimensional space?

How to show that a continuously differentiable function $f:\mathbb{R}^{n}\to \mathbb{R}^m$ can't be a 1-1 when $n>m$? This is an exercise in Spivak's "Calculus on manifolds". I can solve the ...
3
votes
2answers
175 views

Curvature of a regular parametrization

Prove that if $\mu: [a,b] \to \mathbb{R}^n$ is a regular parametrization of a curve then the curvature at $\mu(t)$ is given by: $$\kappa(t) = ...
3
votes
1answer
473 views

Matrix calculus : Find the gradient/derivative?

I know that the derivative of $Tr(Z^TAZ)$ w.r.t $Z$ is $2AZ$. Now I'd like to compute the derivative of $Tr\left[Z^T\left( \operatorname{diag}(ZZ^T\mathbf{1}) - ZZ^T\right)Z\right]$ instead, w.r.t $Z ...
3
votes
2answers
389 views

What strategy do you use when solving vector equations involving $\nabla$?

$\Phi, \Lambda$ are both scalars dependent upon, and $\mathbf u$ is a vector independent of coordinates. I'm trying to express $\Lambda$ in terms from $\mathbf U \cdot \nabla\Lambda = \Phi$ and to ...
2
votes
3answers
408 views

showing / proving curl identity $\nabla \times \left( \frac{1}{r^2} \hat r \right) = 0$

OK, I have to show the following: $$ \nabla \times \left( \frac{1}{r^2} \hat r \right) = 0$$ This should be pretty easy, but I wanted to be sure I was doing this correctly. I set up the matrix: ...
2
votes
1answer
583 views

Proof on showing if F(x,y,z)=0 then product of partial derivatives (evaluated at an assigned coordinate) is -1

The task is as follows: Given: $$F(x,y,z) = 0$$ Goal: Show $\frac{\partial z}{\partial y}$ (evaluated at $x$) * $\frac{\partial y}{\partial x}$ (evaluated at $z$) * $\frac{\partial ...
2
votes
4answers
661 views

What exactly are the “higher order terms” (H.O.T.) in Taylor series expansion in multivariable case?

I know the Taylor series expansion in single variable case: $$ f(x) = f(x_0) + f'(x_0)(x - x_0) + \frac{1}{2}f''(x_0)(x-x_0)^2 + \frac{1}{3!}f^{(3)}(x_0)(x-x_0)^3 + \frac{1}{4!}f^{(4)}(x_0)(x-x_0)^4 ...
2
votes
1answer
263 views

Derivation of weak form for variational problem

My question is about understanding the derivation of the weak form of a variational problem (to be used for the solution via the finite element method). The problem is as follows (it is an image ...
2
votes
1answer
328 views

Implicit function theorem and implicit differentiation

This is perhaps something simple; but I am not quite getting why the implication is true; I seem to be missing something. Supposedly, the implicit function theorem: Let $f: \mathbb{R}^{n + m} ...
2
votes
1answer
139 views

Really Stuck on Partial derivatives question

Ok so im really stuck on a question. It goes: Consider $$u(x,y) = xy \frac {x^2-y^2}{x^2+y^2} $$ for $(x,y)$ $ \neq $ $(0,0)$ and $u(0,0) = 0$. calculate $\frac{\partial u} {\partial x} (x,y)$ and ...
2
votes
2answers
249 views

What does $d_{\textbf{a}} f$ mean?

I have a question regarding the differential $d_{\textbf a} f$. Suppose we have the function $f(x,y)= xy$, and the vectors $\textbf a = (1,1)$ and $\textbf u = (2,1)$. Then, if I understand this ...
1
vote
3answers
1k views

How to show differentiability implies continuity for functions between Euclidean spaces

A function $f: \mathbb{R^n} \to \mathbb{R^m}$ is differentiable at $a$ if there exists a linear map $ \lambda: \mathbb{R^n} \to \mathbb{R^m}$ such that $$\lim_{h \to 0} \frac{\|f(a+h) - f(a) - ...
9
votes
2answers
1k views

Calculate the area on a sphere of the intersection of two spherical caps

Given a sphere of radius $r$ with two spherical caps on it defined by the radii ($a_1$ and $a_2$) of the bases of the spherical caps, given a separation of the two spherical caps by angle $\theta$, ...
7
votes
2answers
101 views

real meaning of divergence and its mathematical intuition

how does divergence which means sink or source equal to ∂Fx/∂x+∂Fy/∂y +∂Fz/∂z.I have been thinking it for a long time and i think "divergence tells us how fast the vector increases when we move apart ...
7
votes
4answers
6k views

L'hospital rule for two variable.

How to use L'hospital rule to compute the limit of the given function $$\lim_{(x,y)\to (0,0)} \frac{x^{2}+y^{2}}{x+y}?$$
6
votes
2answers
119 views

Without Stokes's Theorem - Calculate $\iint_S \operatorname{curl} \mathbf{F} \cdot\; d\mathbf{S}$ for $\mathbf{F} = yz^2\mathbf{i}$

Consider the bounded surface S that is the union of $x^2 + y^2 = 4$ for $−2 \le z \le 2$ and $(4 − z)^2 = x^2 + y^2 $ for $2 \le z \le 4.$ Sketch the surface. Use suitable parametrisations for ...
6
votes
1answer
165 views

Continuous partials at a point without being defined throughout a neighborhood and not differentiable there?

This is a follow-up to Continuous partials at a point but not differentiable there?, but I'll make this question self-contained. Throughout, $f$ will denote a function $\mathbb{R}^2\to\mathbb{R}$. An ...
6
votes
1answer
709 views

Maximum Likelihood Estimation of an Ornstein-Uhlenbeck process

I am wondering whether an analytical expression of the maximum likelihood estimates of an Ornstein-Uhlenbeck process is available. The setup is the following: Consider a one-dimensional ...
5
votes
4answers
138 views

How to find the minimum value of this function?

How to find the minimum value of $$\frac{x}{3y^2+3z^2+3yz+1}+\frac{y}{3x^2+3z^2+3xz+1}+\frac{z}{3x^2+3y^2+3xy+1}$$,where $x,y,z\geq 0$ and $x+y+z=1$. It seems to be hard if we use calculus methods. ...
5
votes
5answers
105 views

How to prove that $\lim_{(x,y) \to (0,0)} \frac{x^3y}{x^4+y^2} = 0?$

How to prove that $\lim_{(x,y) \to (0,0)} \dfrac{x^3y}{x^4+y^2} = 0?$ First I tried to contradict by using $y = mx$ , but I found that the limit exists. Secondly I tried to use polar ...
5
votes
2answers
231 views

Why is arc length not a differential form?

I read that the arc length is not a differential form. But I don't understand why it isn't. I understand that differential forms are integrands and arc length is an expression which is integrable. ...
5
votes
2answers
239 views

A little help integrating this torus?

Let $\mathbf{F}\colon \mathbb{R}^3 \rightarrow \mathbb{R}^3$ be given by $$\mathbf{F}(x,y,z)=(x,y,z).$$ Evaluate $$\iint\limits_S \mathbf{F}\cdot dS$$ where $S$ is the surface of the torus ...
5
votes
1answer
512 views

Volume of an n-simplex

It's rather tedious to show using Fubini's Theorem and induction on $n$ that the volume of the region $x_1+x_2+...+x_n \leq 1$ with $x_1,...,x_n$ nonnegative is $\frac{1}{n!}$. Is there an easier way ...
5
votes
1answer
1k views

The Gradient as a Row vs. Column Vector

Kaplan's Advanced Calculus defines the gradient of a function $f:\mathbb{R^n} \rightarrow \mathbb{R}$ as the $1 \times n$ row vector whose entries respectively contain the $n$ partial derivatives of ...