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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12
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5answers
35k views

Surface Element in Spherical Coordinates

In spherical polars, $$x=r\cos(\phi)\sin(\theta)$$ $$y=r\sin(\phi)\sin(\theta)$$ $$z=r\cos(\theta)$$ I want to work out an integral over the surface of a sphere - ie $r$ constant. I'm able to derive ...
14
votes
1answer
298 views

Mathematical meaning of certain integrals in physics

While studying on texts of physics I notice that differentiation under the integral sign is usually introduced without any comment on the conditions permitting to do so. In that case, I take care of ...
15
votes
1answer
2k 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 ...
13
votes
1answer
768 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 ...
9
votes
2answers
577 views

Is there a discontinuous function on the plane having partial derivatives of all orders?

If one requires simply the existence of partial derivatives of first order rather than all orders, then a standard example is the function $$ f(x,y) = \left\{\begin{array}{l l} \frac{2xy}{x^2+y^2} ...
8
votes
1answer
1k views

If derivative of a function is the zero function in $\mathbb R^n$, then the function is constant when the domain is path-connected

Some definitions first. Let $A \subseteq \mathbb R^n$. Let $x,y \in A$. A path between $x$ and $y$ is a continuous function $f: [0,1] \rightarrow \mathbb{R}^n$ with $f(0) = x$ and $f(1) = y$. The set ...
10
votes
3answers
4k views

Understanding the concept behind the Lagrangian multiplier

I've been trying to understand the principles behind the Lagrangian multipliers and I think I've got a rough understanding of it. Would appreciate it if you guys could help me answer a few questions! ...
9
votes
1answer
138 views

dropping injectivity from multivariable change of variables

The change of variables for multivariable integration in Euclidean space is almost always stated for a $C^1$ diffeomorphism $\phi$, giving the familiar equation (for continuous $f$, say) ...
3
votes
2answers
3k views

On the absolute value of Jacobian determinant - variable transformation in multi-integral

I would like to change some variables in a integral and encountered to an issue. I create here 2 simple examples to describe my questions: Exp 1. Suppose we want to change $(x,y)$ to $(u,v)$ such ...
22
votes
5answers
2k views

Geometric interpretation of mixed partial derivatives?

I'm looking for a geometric interpretation of this theorem: My book doesn't give any kind of explanation of it. Again, I'm not looking for a proof - I'm looking for a geometric interpretation. ...
6
votes
0answers
219 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 ...
6
votes
5answers
2k views

Reference for multivariable calculus

I'm looking for a book to learn multivariable calculus that is rigorous, but not overly technical, and also provides meaningful insight. Standard calculus texts like Stewart and Thomas are too ...
4
votes
1answer
681 views

On finding polynomials that approximate a function and its derivative (extensions of Stone-Weierstrass?)

The Stone-Weierstrass Theorem tells us that we can approximate any continuous $f:\mathbb{R}^n\to\mathbb{R}$ arbitrary well on a compact subset of $\mathbb{R}^n$ by some polynomial. Suppose that $f$ is ...
13
votes
4answers
16k views

L'hospital rule for two variable.

How to use L'hospital rule to compute the limit of the given function $$\lim_{(x,y)\to (0,0)} \frac{x^{2}+y^{2}}{x+y}?$$
11
votes
5answers
6k views

Multivariable Calculus books similar to “Advanced Calculus of Several Variables” by C.H. Edwards

I am currently trying to teach myself multivariable calculus using C.H. Edwards' "Advanced Calculus of Several Variables", but the text unfortunately doesn't have very many problems with solutions. ...
9
votes
2answers
847 views

Partition of Unity in Spivak's Calculus on Manifolds

I have a question about partitions of unity specifically in the book Calculus on Manifolds by Spivak. In case 1 for the proof of existence of partition of unity, why is there a need for the function ...
9
votes
5answers
976 views

Distinction between vectors and points

I've been wondering for some time now about the difference between a point and a vector. In high school, it was very important to distinguish them from each other, and we used the notation $(x,y,z)$ ...
8
votes
3answers
925 views

Newton's method in higher dimensions explained

I'm studying about Newton's method and I get the single dimension case perfectly, but the multidimensional version makes me ask question... In Wikipedia Newton's method in higher dimensions is ...
8
votes
1answer
1k views

Taylor's theorem in Banach spaces

Let $f$ be a real function of a single real variable. Suppose that $f$ is $n$ times differentiable at some $x$, for some integer $n\geq 1$. Making no further assumptions, we have $$ f(x+h) = f(x) + ...
8
votes
2answers
2k views

Can “being differentiable” imply “having continuous partial derivatives”?

Consider the following theorem: Let $E$ be a subset of ${\bf R}^n$, $f:E\to {\bf R}^m$ be a function, $F$ be a subset of $E$, and $x_0$ be an interior point of $F$. If all the partial derivatives ...
6
votes
2answers
576 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
1answer
322 views

If $f$ is twice differentiable, $(f(y) - f(x))/(y-x)$ is is differentiable

Suppose $f: \mathbb{R} \to \mathbb{R}$ is a $C^{1}$ function. Then, define a new function $F: \mathbb{R}^{2} \to \mathbb{R}$ by: $$ F(x,y) = \begin{cases} \displaystyle \frac{f(y) - f(x)}{y - x} ...
4
votes
1answer
234 views

Limit with integral or is this function continuous?

Hello I need to show one identity and one limit. I am having problems with it. notation: $x_i$ is i-th coordinate of $x$ $B(x,r)$ ball with center $x$ and radius $r$ $S(x,r)$ sphere with center ...
4
votes
3answers
440 views

correcting a mistake in Spivak

Spivak's Calculus on Manifolds asks the reader to prove this (problem 1-8, pp.4-5): If there is a basis $x_1, x_2, ..., x_n$ of $\mathbb{R}^n$ and numbers $\lambda_1, \lambda_2, ..., \lambda_n$ ...
3
votes
2answers
662 views

A curve parametrized by arc length

Let $C$ be a plane curve parametrized by arc length by $\alpha(s)$, $T(s)$ (unit tangent vector) and $N(s)$ (unit normal vector). Prove that $$\frac{d}{ds} N(s)=-\kappa(s)T(s).$$ I know that ...
2
votes
1answer
8k views

How to tell if a limit of a multi-variable function exists?

Since I began studying limits of multi-variable functions, I have been baffled with this question: how can one tells if a limit exists or not? I don't know if it's the right way to solve this kind of ...
11
votes
3answers
376 views

Determine whether $\lim_{(x,y)\to (2,-2)} \dfrac{\sin(x+y)}{x+y}$ exists.

I am trying to determine whether $\lim_{(x,y)\to (2,-2)} \dfrac{\sin(x+y)}{x+y}$ exists. I should be able to use the following definition for a limit of a function of two variables: Let $f$ be a ...
11
votes
1answer
682 views

Problem 3-38 in Spivak´s Calculus on Manifolds

This is not homework. Problem 3-38 reads: Let $A_{n}$ be a closed set contained in $(n,n+1)$. Suppose that $f:\mathbb{R}\rightarrow \mathbb{R}$ satisfies $\int_{A_{n}}f=(-1)^{n}/n$ and $f(x)=0$ for ...
6
votes
1answer
152 views

$ \int_1^2\int_1^2 \int_1^2 \int_1^2 \frac{x_1+x_2+x_3-x_4}{x_1+x_2+x_3+x_4}dx_1dx_2dx_3dx_4 $

Evaluate $$I= \int_1^2\int_1^2 \int_1^2 \int_1^2 \frac{x_1+x_2+x_3-x_4}{x_1+x_2+x_3+x_4}dx_1dx_2dx_3dx_4$$ Answer Options: $1$ $\frac{1}{2}$ $\frac{1}{3}$ $\frac{1}{4}$ I need some ...
6
votes
2answers
1k views

Equivalent condition for differentiability on partial derivatives

I want to extend the concept of derivative of a real function of real variable to a function $f:A\subset \mathbb{R}^n \to \mathbb{R}^m$ with $A$ open. If $x_0 \in A$ then I say that $f$ has derivative ...
5
votes
5answers
294 views

Interpreting higher order differentials

I'm trying to understand Taylor's Theorem for functions of $n$ variables, but all this higher dimensionality is causing me trouble. One of my problems is understanding the higher order differentials. ...
5
votes
1answer
1k views

Continuity of one partial derivative implies differentiability

Let $f: \mathbb{R}^2 \rightarrow \mathbb{R}$ be a function such that the partial derivatives with respect to $x$ and $y$ exist and one of them is continuous. Prove that $f$ is differentiable.
5
votes
1answer
543 views

Use Stokes's Theorem to show $\oint_{C} y ~dx + z ~dy + x ~dz = \sqrt{3} \pi a^2$

I am a little stuck on the following problem: Use Stokes's Theorem to show that $$\oint_{C} y ~dx + z ~dy + x ~dz = \sqrt{3} \pi a^2,$$ where $C$ is the suitably oriented intersection of the ...
4
votes
1answer
1k views

if the curvature is constant and positive, then it is on the circunference

I'm trying to prove that if $\alpha(t)=(x(t),y(t))$ is a $C^2$ regular curve $(\alpha'\neq0)$ with constant and positive curvature, then $\alpha$ is on the circunference and if $\alpha$ is the ...
2
votes
2answers
157 views

$f:\mathbb R^{2} \rightarrow \mathbb R$ s.t ${f(x,y)}={{xy}\over {x^{2}+y}}$ is not continuous at the origin

$f:\mathbb R^{2} \rightarrow \mathbb R$ is defined as $${f(x,y)}={{xy}\over {x^{2}+y}}$$; when $x^{2}+y\neq 0$ and $$f(x,y)=0$$ otherwise. To show this is not continuous at the origin . ...
2
votes
2answers
101 views

Use implicit function theorem to show $O(n)$ is a manifold

In class today our teacher mentioned that one can use the implicit function theorem to show that $O(n) \subseteq \mathbb{R}^{n^2}$ is a submanifold...that is, map $A \mapsto A^* A$, and set it equal ...
2
votes
2answers
289 views

Using nabla with partial derivatives and the Laplace operation $\partial_x^2+\partial_y^2+\partial_z^2$

Source of the problem p.812 here. Suppose $$\bar{F}(x,y,z)=(xy-z^2)\bar{i}+(xyz)\bar{j}+(x-y^2-z^2)\bar{k}.$$ I am concerned where I need to nabla an unit vector for example with $$\triangledown ...
1
vote
3answers
112 views

Demand $z=x+y$ and $x^2/4 + y^2/5 + z^2/25 = 1$. What is the maximum value of $f(x,y,z) = x^2+y^2+z^2$?

Demand $z=x+y$ and $x^2/4 + y^2/5 + z^2/25 = 1$. What is the maximum value of $f(x,y,z) = x^2+y^2+z^2$? I've been attempting this with Lagrange multipliers in a few different ways. However, the ...
11
votes
4answers
2k views

Is the max of two differentiable functions differentiable?

Given that $f$ and $g$ are two real functions and both are differentiable, is it true to say that $h=\max{(f,g)} $ is differentiable too? Thanks
10
votes
4answers
4k views

Del. $\partial, \delta, \nabla $: Correct enunciation

I've come across various different symbols being pronounced as "del". What is the internationally accepted del? If not internationally, then what's the English/American(specify which one if they are ...
9
votes
1answer
5k views

Solving quadratic vector equation

Hope it is a right place to ask how to solve the equation on $\mathbf x$: $$ \mathbf x^T \mathbf A\mathbf x + \mathbf x^T \mathbf b + c = 0. $$ where: $\mathbf x$ is an $n\times 1$ column vector ...
8
votes
2answers
321 views

Multivariable limit with logarithm

I have to prove that the limit $$\lim_ {{(x,y)} \to {(0,0)}} \frac{xy^2\ln\frac{|x|}{|y|}}{{(x^2+y^2)}^{\frac 12}}$$ does not exist. I've tried to find two different paths that show that the limit ...
7
votes
1answer
371 views

Helix in a helix

I am trying to work out a "helix in a helix" mathematically. Intuitively I think of this as a steel cable, which is made up of a number of smaller steel cables all bound together in spiral. If I ...
6
votes
2answers
239 views

Is this a correct use of the squeeze theorem?

I have to find the following limit: $$ \lim_{x\to0,y\to0} \frac{x^2y^2}{x^2+y^4}=[\frac{0}{0}] $$ I try to reach the origin moving on the y-axis ($x=0$): $$ \lim_{y\to0} \frac{0}{y^4}=0 $$ I get ...
6
votes
2answers
612 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 ...
6
votes
3answers
744 views

Proving that the function $\frac{x^2y}{x^2 + y^2}$ 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 $(0,0$)?
5
votes
1answer
147 views

Can the following triple integral be computed via elementary calculus methods?

Consider the following triple integral: $$\int_0^{2\pi}\int_0^1 \int_0^1 xy\sqrt{x^2 + y^2 -2xy\cos(\theta)} \, dx \, dy \, d\theta$$ A solution was provided to this integral by Jack D'Aurizio ...
5
votes
2answers
836 views

Is the boundary of the unit sphere in every normed vector space compact?

I wanted to ask whether the boundary of the unit sphere in every normed vector space is compact? I know that this is true for simple examples, but how is it in general?
4
votes
2answers
284 views

Understanding the derivative as a linear transformation

It's been a while now I am studying multivariable calculus and the concept of differentiation in space (or higher dimension). I saw relative posts but one question remains. I can't understand the ...
4
votes
5answers
2k views

What is the intuition behind the unit normal vector being the derivative of the unit tangent vector?

I've seen the math, but... It just doesn't make sense to me. How is the slope going to point perpendicular to the vector that is clearly a straight line not going in that direction?