Questions tagged [multivariable-calculus]

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).

6,128 questions with no upvoted or accepted answers
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20
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499 views

What can be said about the level set of the real part of an analytic function?

I am working with a function $F(z;a)$, for $z\in \mathbb{C}$ and $a$ being a set of parameters, from which I need to analyze the level set $\text{Re}(F(z))=0$ (for a fixed set of parameters $a$, which ...
13
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710 views

Lagrange multipliers in Calculus of Variations

I am trying to learn about Calculus of Variations and I am beginning to see some constrained optimization problems in the domain of functionals, by using Lagrange multipliers. It seems that things ...
11
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0answers
232 views

Tricky surface integral of vector field

We have the embedded surface $S= \{(x,y,z)\in \mathbb{R}: z = e^{1-(x^2 + y^2)^2}, z>1\}$ and the vector field $\mathbf F:\mathbb{R}^3\to \mathbb{R}^3; (x,y,z)\mapsto (x e^{y^2}, 2ye^{x^2}, 5-3z) $....
11
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0answers
345 views

What vector field property means “is the curl of another vector field?”

I know that a vector field $\mathbf{F}$ is called irrotational if $\nabla \times \mathbf{F} = \mathbf{0}$ and conservative if there exists a function $g$ such that $\nabla g = \mathbf{F}$. Under ...
11
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1answer
372 views

Can a differential k-form be integrated on a manifold that is not k-dimensional?

For example, can you integrate a 2-form on some curve, a 1-dimensional manifold, or some 3-dimensional manifold? I know that Stokes's Theorem states that if you integrate $\omega \in \mathcal A^{k-1}(...
11
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1answer
821 views

What is the best way to learn Differential forms?

I'm taking a Multivariable Calculus class and my teacher has just started Differential forms. It is not making a lot of sense, though. I have tried reading "Geometric Approach to Differential forms" ...
10
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0answers
281 views

When is $\int_0^1 \int_0^1 \frac{f(x) - f(y)}{x-y} \, \text{d} x \, \text{d} y = 2 \int_0^1 f(t) \log\left(\frac{t}{1-t}\right) \, \mathrm{d} t$?

Double integrals of this type sometimes appear when using differentiation under the integral sign with respect to two variables. Therefore, I am interested in reducing them to (simpler) single ...
10
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0answers
120 views

How weird can the boundary be so that the fundamental theorems of vector calculus hold?

Let $\Omega$ be a connected open set in $\Bbb R^n$. Suppose that I want theorems in multivariables calculus like divergence theorem or its relative like Green's identities or even Stoke's theorem to ...
10
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1answer
151 views

Double integral - transformation

I'm trying to calculate $$\iint_{\Omega } e^{(x+y^2)^{3/2}} \,\mathrm{d}A,$$ where $$\Omega =\{x,y>0 : x+y\leq 2\}. $$ Not sure where to go with it. I need to find a transformation and then ...
10
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775 views

Nasty Integral - Closed form solution?

Any suggestions on how to integrate this beast?: $$\int_0^{\omega_t}\int_{\omega_t}^f\sin^2(\omega_{12}/2)\sin^2(\omega_{23}/2)d\omega_{23}d\omega_{12}$$ where: $f{} = 2\pi+2\tan^{-1}(y,x)$ $y = -...
10
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1answer
2k views

Singular jacobian matrix?

I have a series of questions, in various degrees of befuddled muddledness (and they are related to my previous questions: this and this) First question: how do I do a change of variable if the ...
9
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1answer
235 views

Evaluating the Surface Integral $\iint_{x^3+y^3+z^3=a^3} \frac{\bf{x}}{||\bf{x}||} \cdot d\bf{S}$

Compute the surface integral: $$\int_S({x\over \sqrt{x^2+y^2+z^2}}, {y\over \sqrt{ x^2+y^2+z^2}}, {z\over \sqrt{x^2+y^2+z^2}}) \cdot \vec n \ dS$$ where $S: x^3+y^3+z^3=a^3$ The first ...
9
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2k views

How to take limit *along a path*

So in multivariable calculus for a limit of a function to exist, the limits of the function along all possible paths must exist and equal the same value. But how does one calculate the limit along a ...
9
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650 views

History of line integral.

I'm looking for some information about how the line integral was discovered, since I've been looking for a long time for this. I found that Riemann could integer discontinuity functions, then Poisson ...
8
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1answer
162 views

Normal orthant probability in six variables

Let $\mathbf{X} \sim N(\mathbf{0}, \mathbf{\Sigma})$ be a $6$-dimensional Gaussian vector with covariance matrix of the form $$\mathbf{\Sigma} = \begin{pmatrix} 1 & c \\ c & 1 \end{pmatrix} \...
8
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670 views

Let $f: A \to \mathbb{R^n}$ be of class $C^r$; $Df(x)$ is non-singular for $x\in A$. Show that even if $f$ is not 1-1, the set $B=f(A)$ is open.

Let $A$ be open in $\mathbb{R^n}$; let $f: A \to \mathbb{R^n}$ be of class $C^r$; assume $Df(x)$ is non-singular for $x\in A$. Show that even if $f$ is not one-to-one on $A$, the set $B=f(A)$ is open ...
7
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1answer
109 views

Can a spiral have its centroid at the origin?

A spiral is a curve $\gamma$ with the polar equation $r=f(\theta)$ where $f$ is a continuous positive strictly monotone function on some interval $[a, b]$, $-\infty<a<b<\infty$. Best known ...
7
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1k views

Rudin's Rank theorem

Rudin states the following: 9.32 Theorem: Suppose $m,n,$ are nonnegative integers, $m\geq r,n\geq r$, $F$ is a $C^1$ mapping of an open set $E\subset R^n$ into $R^m$, and $F'(x)$ has rank $r$ for ...
7
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238 views

Euler-Lagrange Equation and “Eigen Value ”

The Eigen value $\lambda(t)$ which is characterised by the Rayleigh quotient (where $t$ is a scalar variable): $$R(u,\Omega_t)= \frac{\int_{\Omega_t} |\nabla u|^2 dy }{\int_{\Omega_t} u^2 dy}$$ $\...
7
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294 views

Multivariable Integral, How to compute it?

Can anybody please tell me, how to evaluate a multivariate integral with a gaussian weight function. $$ \mathcal{Z_{n}} \equiv\int_{-\infty}^{\infty} \exp\left(-a\sum_{j = 1}^{n}x_{j}^2\right)\, {\rm ...
7
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0answers
172 views

Question on using Leibniz formula to derive thin-film equation from Navier-Stokes

I am trying to work through the derivation in this paper by Petr Vita, which derives a thin-film simplification of the Navier-Stokes equation, similar to the Reynolds or Lubrication Equation, but ...
7
votes
1answer
748 views

Surface integral for a scalar function defined on a discrete surface

Imagine a polyhedral, discrete surface embedded in $\mathbb{R}^3$. Its faces are all triangles. For each vertex, one can compute the discrete mean and Gaussian curvatures and evaluate the sum of ...
7
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203 views

Proof of inverse function theorem by approximation property

In proving the inverse function theorem using the approximation characterization of the derivative, we are given $F:\mathbb{R}^n \to \mathbb{R}^n$ such that  $$F(p_0 + h) - F(p_0) = DF_{p_0}(h) + o(\|...
6
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0answers
75 views

Smooth submanifold of $\mathbb R^6$. Not smooth submanifold of $\mathbb R^3$

So I'm pretty new to studying manifolds and have little to no background on differential geometry, but this is a question from lecture notes on a multivariable analysis unit: Show that $S:=\{(x^2,y^...
6
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1answer
109 views

All smooth real functions are related by coordinate change?

Suppose you have two smooth functions $F,G:\mathbb{R}^n\rightarrow \mathbb{R}$ with the property that none of the partial derivatives vanish anywhere. I wonder if it's possible to find functions $P:\...
6
votes
1answer
64 views

Partial derivative notation

Suppose $F(x,y,z)=x+y+z$. Then the partial derivative of $F$ w.r.t. $x$ is $1$. (Most books don't mention that $y$ and $z$ are held constants and it's sort of implied.) But what if $z$ was a function ...
6
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219 views

Does the converse of the chain rule hold in general?

I've found the following theorem for single variable real valued functions: The converse of the chain rule: Suppose that $f,g$ and $u$ are related so that $f(x)=g(u(x))$. If $u$ is continuous at $...
6
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0answers
582 views

Volume of n-dimensional ball in L1 norm with change of variables

For a homework problem, I need to find a recursive equation that relates the volume of an $n$-dimensional ball $V_n(r)$ of radius $r$ to that of an $(n-2)$-dimensional ball, expressed by $V_{n-2}(r)$. ...
6
votes
1answer
49 views

Is there an easier way to show that a tetrahedron is optimal?

Basically, I'm trying to address a problem that somewhat mirrors electron geometry, and I'm phrasing it like this: Consider four points in three dimensional space ($P_1,P_2,P_3,P_4$) such that each $...
6
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0answers
2k views

Why is the negative of the gradient the direction of greatest descent?

I imagine it as if one is going up a physical hill. It doesn't seem like there's a guarantee that going in the opposite direction of greatest increase in height will necessarily be the direction of ...
6
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0answers
77 views

Reference for the fact that a smooth function analytic on every line is itself analytic

Let $f \in \mathcal C^\infty(\mathbb R^p)$ ($p \geq 2$) be a smooth function such that the functions $g_d(t) := f(td)$ are all analytic for all $t \in \mathbb R$ and all $d \in \mathbb R^p.$ (i.e. $f$ ...
6
votes
1answer
441 views

Understanding the setup for the probability that $Ax^2+Bx+C$ has real roots if A, B, and C are random variables uniformly distributed over (0,1).

Suppose that $A, B,$ and $C$ are independent random variables, each being uniformly distributed over $(0,1)$. What is the probability that $Ax^2 + Bx + C$ has real roots? First, I set $P(B^2 - 4AC \...
6
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0answers
259 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 |^{3}}\right]d^{3}\...
6
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0answers
270 views

Algorithm to calculate multiple integral.

One of the major difficulties of student in advanced calculus (including myself when student) is to obtain the extremes of repeated integrals to calculate the volume integral in $R^n$ i.e. transform ...
6
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0answers
4k views

Good introductory book for matrix calculus

Hi I am an electronics graduate and working on image processing for the past one year...I have a basic exposure to linear algebra(thanks to Gilbert Strang..!!!). Now I am facing problems with matrix ...
6
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0answers
333 views

Behaviour at infinity of a function in terms of first and second derivatives

In a paper (dealing with spectra of certain Schrodinger operators) I found the following assumption for a function $f\in C^\infty(\mathbb R^n;\mathbb R)$: there exists a constant $C>0$ and a ...
5
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62 views

Surface Area of $n$-Dimensional Sphere with Multiple Hyperplanar Cuts

Let $S^{n-1}\subset\mathbb{R}^n$ be the unit sphere, and $v_1, \cdots, v_m\in S^{n-1}$ be $n$-dimensional unit vectors. Each of these vectors defines an $n-1$ dimensional hyperplane, which cuts $S^{n-...
5
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0answers
119 views

Compute the derivative $\partial_t\int_{\{u(t,\cdot) >0\} } 1\, dx$ in the sense of distributions

Let $u:\Omega\subset \mathbb R^N \to \mathbb R$ be bounded function that solves (in the sense of distributions) an evolution PDE $\partial_t u(t,x)= L(u(t,\cdot))(x)$, where $L$ is some elliptic ...
5
votes
1answer
141 views

Intuition of divergence and curl

There is the well known expression for the divergence of a vector field $V$ as the limit of smaller and smaller surfaces of the flux of a surface. However it occurred to me that there is another way ...
5
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0answers
55 views

Proving the determinant of the derivative of a function can not be zero on any open set.

I'm working on the following problem for my multivariable calculus course: Let $f: R^2 \to R^2 $ be a $C^1$ function such that for each $y \in R^2$, the set $f^{-1}(y)$ is finite. Show that $\det Df(...
5
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0answers
126 views

Triangle inequality for integrals, but for an arbitrary norm

Given an arbitrary norm on $\mathbb{R}^q$. For a continous function $f: [a,b]\times\mathbb{R}^q \rightarrow \mathbb{R}^q$, I want to find out whether $$ \left\|\int_{a}^{b} f(x,y_1,...,y_q)\,\mathrm{...
5
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0answers
121 views

Positive-Eigenvalue Jacobian $\Rightarrow$ Invertible?

Let $f : \mathbb{R}^n \to \mathbb{R}^n$ have Jacobian $J\,f : \mathbb{R}^n\to\mathsf{M}_n(\mathbb{R})$ which has positive eigenvalues everywhere. Is $f$ (globally) invertible on it's range/injective? ...
5
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0answers
124 views

An annoying optimization problem

At first I thought the following problem looked simple, but I've had serious problems pinning it down: Suppose that $\prod_{i=1}^k x_i$ is fixed. Then find the minimum value of $$\sum_{i=1}^k (1 - ...
5
votes
1answer
94 views

Diagram of the tangent vector at a point along a curve on a manifold in generalized coordinate system

The interpretation of vectors as partial derivatives of the position vector and dual vectors as stacks is visually very appealing. I understand that in the book Gravitation by Misner the passage from ...
5
votes
1answer
195 views

Explanation of “without loss of generality” in an application of Inverse Function Theorem.

Let $U$ be an open subset of $\mathbb R^{n+m}=\mathbb R^n\times \mathbb R^m$ and $g:U\to\mathbb R^m$ a $C^1$ function. Let $p=(x_0,y_0)\in U$ be a point such that $$g'(p):\mathbb R^{n+m}\to \mathbb R^...
5
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0answers
98 views

Discontinuous function which is continuous when restricted to any algebraic curve

Background: In a first course on multivariable calculus, it's really common to find examples of functions which are discontinuous, but continuous when restricted to any line, in order to build ...
5
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0answers
59 views

Integrate over sinusoidal region $\int\int(1 - \cos 2\theta ) (1 + \cos(\frac{\theta + 2 \pi u}2 )) du \,d\theta$

I'm trying to obtain a closed form expression for the following definite integral in $\mathbb{R}^2$: $$ \int_{0}^{\pi} \int_{u = 0}^{u= A \sin\theta} f(\theta, u)\,\mathrm{d}u \,\mathrm{d}\theta ~,\...
5
votes
0answers
384 views

Are there any simple examples of Kolmogorov-Arnold representation?

I had never heard of the Kolmogorov-Arnold Representation Theorem before. It states roughly that any multivariable function can be represented by repeatedly adding a single variable function whose ...
5
votes
0answers
99 views

Show that $f+\epsilon g\in{\rm Diff^1}(\Bbb R^m)$ for all $\epsilon\in(-\epsilon_0,\epsilon_0)$

This is an exercise on page 220 of Analysis II of Amann and Escher Here $\rm Diff^k$ means the set of diffeomorphisms where the $k$-th derivative is also an homeomorphism. My work below. The ...
5
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0answers
140 views

Is there a set of “canonical” partial derivative exercises that “tell the whole story”?

I find the field of partial differentiation of complicated multivariable expressions to have many subtleties which lead to unexpected results, or to my applying operations incorrectly. I'm wondering ...