1
vote
1answer
26 views

Understanding arguments to functions in $\mathbb R^n$

Example of two theorems I have problems with: Mean value theorem: $U\subseteq\mathbb R^n$ open, $f:U\to\mathbb R^m$continuously differentiable, $x\in U$, $\xi\in\mathbb R^n$ such that $x+t\xi\in U$ ...
2
votes
2answers
28 views

In regards to lagrange multipliers, Confusion about derivation.

In my calculus III textbook, the following sentence is causing trouble for me and preventing me from understanding the theory behind Lagrange multipliers. "Since the gradient vector for a given ...
2
votes
0answers
38 views

General formula for dependent probability distributions

Recently I encountered the following problem: What is the mean distance between two random points on a unit square? I understand pen and paper methods for solving this exist however I'm ...
0
votes
1answer
18 views

What is an intuitive extension of extreme-values and critical points in one variable to multiple variables?

While it is simple to grasp limits in multiple variables, since the formal definition extends in the obvious way, I am having a harder time grasping the same concept with critical points and extreme ...
3
votes
3answers
154 views

Line Integrals and Surface Integrals

Can someone please explain what surface integrals and line integrals are measuring? Is a line integral the arc length along a surface, and a surface integral is the surface area? Also, why is a line ...
12
votes
10answers
410 views

Why is it that $\int_a^b \int_c^d f(x)g(y)\,dy\,dx=\int_a^b f(x)\,dx \int_c^d g(y)\,dy$?

The title sums it up. It's simple to prove, but I'm wondering if there is a geometric interpretation?
1
vote
1answer
38 views

If $z = f(x, y)$, then why are $\partial_x z$ and $\partial_y z$ functions of x and y also? [Stewart P905]

This is Figure 5 from P905 which appears to show this, but Stewart doesn't write this explicitly or explain. I'm interested in an informal, intuitive explanation please. I'm not interested in a ...
6
votes
1answer
142 views

Stokes' Theorem Explanation

Can someone explain what Stokes' Theorem is measuring? What would taking the integral of a vector on a surface give you? When would you use it? This is the only definition I have and I don't really ...
2
votes
2answers
74 views

Gradient and Swiftest Ascent

I want to understand intuitively why it is that the gradient gives the direction of steepest ascent. (I will consider the case of $f:\mathbb{R}^2\to\mathbb{R}$) The standard proof is to note that the ...
3
votes
2answers
97 views

Meaning of Normal Vector in Surface Integration

Is there a good interpretation of what the normal vector (and its magnitude) $$\mathbf{N}=\frac{\partial \mathbf{X}}{\partial s}\times\frac{\partial\mathbf{X}}{\partial t}$$ to the parametric surface ...
3
votes
2answers
91 views

Show that $f(x,y)= \|x-y\|_2^2$ is differentiable

Problem: Show that $f: \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}$ with $f(x,y)=\|x-y\|_2^2$ is differentiable and compute its differential at every point in the domain of $f$Note: $\| \cdot ...
2
votes
3answers
260 views

Polar coordinates for $xz$-plane: $z = r\sin\theta$ ? [Stewart P1091 16.7.25]

$1.$ The unit disk is projected onto the xz-plane, so shouldn’t $x = 1\cos \theta$ and $\color{red}{z = 1 \sin \theta} $? User Semsem below kindly identified the problem: The normal to the ...
7
votes
2answers
136 views

How would one arrive at the formulas for divergence and curl?

It has been some years since I've taken multivariable calculus now, but there's something I really never understood: how people would discover the expressions for divergence and curl. I mean, the ...
7
votes
1answer
111 views

How to change variables in a surface integral without parametrizing

This is a doubt that I carry since my PDE classes. Some background (skippable): In the multivariable calculus course at my university we made all sorts of standard calculations involving surface ...
2
votes
1answer
80 views

What are all the boundary curves for this combined cone and cylinder? [2013 10C]

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 ...
2
votes
1answer
72 views

Finding $g_i:\mathbb{R}^n\to\mathbb R$ s.t $f(x)=\sum\limits_{i=1}^nx_i\cdot g_i(x)$

Let $f:\mathbb R^n\to\mathbb R$ differntiable and $f(0)=0$. Prove exist $g_i$ s.t for $x=(x_1,\dots,x_n):f(x)=\sum\limits_{i=1}^nx_i\cdot g_i(x)$. hint:$f(x)=\int\limits_0^1f\prime(tx)dt$. I dont ...
0
votes
2answers
103 views

Intuition behind $\vec{e_i} \times \vec{e_j}=\epsilon_{ijk} \vec{e_k}$ (Levi Civita)

Let $\vec{e_i}$ denote a unit vector. Then we can write: $\vec{e_i} \times \vec{e_j}=\epsilon_{ijk} \vec{e_k}$, where $\epsilon_{ijk}$ is the Levi Civita symbol. Can someone intuitively explain me ...
1
vote
1answer
365 views

Visual intuition partial/directional derivative

I've had some trouble with the (visual) intuition behind the directional derivatives so I decided to take a step back and look up the visual intuition behind partial derivatives, which I think I do ...
2
votes
1answer
83 views

smooth approximate parameterization to polygonal boundary

I can "almost" parameterize the boundary of a square using $${\bf r}(t) = (\cos t)^{1/p} {\bf i} + (\sin t)^{1/p} {\bf j},$$ $0\leq t\leq 2 \pi$, and $p$ is odd. This parameterization is smooth (or at ...
1
vote
1answer
331 views

What is the intuition behind the Lagrange multiplier?

I know that the minimum or maximum point is achieved when the gradient in the constraint function is parallel to the gradient on the $f$ function. But why the Lambda is called the Lagrange ...
27
votes
4answers
1k views

The Meaning of the Fundamental Theorem of Calculus

I am currently taking an advanced Calculus class in college, and we are studying generalizations of the FTC. We just started on the version for Line Integrals, and one can see the explicit symmetry ...
6
votes
1answer
178 views

Intuition behind $\nabla \times \mathbf{F}$

Is there a simple explanation why this form for the curl of a vector field $\mathbf{F}$, $$\nabla \times \mathbf{F}=\begin{vmatrix} \hat{x} & \hat{y} &\hat{z} \\ \frac{\partial}{\partial ...
9
votes
1answer
906 views

What is the intuition behind the Wirtinger derivatives?

The Wirtinger differential operators are introduced in complex analysis to simplify differentiation in complex variables. Most textbooks introduce them as if it were a natural thing to do. However, I ...
10
votes
5answers
657 views

Intuition on the curl formula

I'm trying to understand the formula for the curl: $$\operatorname{Curl}(F) = \nabla \times F $$ Why the vector product? What means the vector product of a vector and a operator? What is the meaning ...
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 ...
3
votes
3answers
1k views

Intuitive proof of multivariable changing of variables formula (jacobian) without using mapping and/or measure theory?

iWhat is a intuitive proof of multivariable changing of variables formula (jacobian) without using mapping and/or measure theory? I was thinking that textbooks make the proofs over complicate. If ...
6
votes
0answers
241 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 ...
9
votes
1answer
483 views

Information captured by differential forms

My advanced calculus class is currently doing differential forms and I have a hard time really understanding what they are all about. I can read the proofs of the theorems given in Rudin's PMA chapter ...
2
votes
0answers
68 views

Intuition about moment function derivation [OR] derivatives involving a time varying integration domain

$$ m_{{pq}}(t)=\iint\limits_{R(t)}h(x,y) dx dy $$ where $ R(t)$ the domain of integration is time varying (In fact it is the only one which is time varying). And $$ h(x,y) = x^p y^q f(x,y) dx dy ...
32
votes
5answers
4k views

Intuitive interpretation of the Laplacian

Just as the gradient is "the direction of steepest ascent", and the divergence is "amount of stuff created at a point", is there a nice interpretation of the Laplacian (a.k.a. divergence of gradient)? ...
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 ...
17
votes
6answers
5k views

What is the 'implicit function theorem'?

Please give me an intuitive explanation of 'implicit function theorem'. I read some bits and pieces of information from some textbook, but they look too confusing, especially I do not understand why ...
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} = ...