Linked Questions

2
votes
1answer
6k views

Jacobian physical meaning [duplicate]

Possible Duplicate: What is Jacobian Matrix? Is there any physical intuition for the Jacobian? I understand that it is the matrix of partial derivatives and how to construct it. What I want to ...
0
votes
3answers
785 views

What is a Jacobian? [duplicate]

I understand how to calculate the Jacobian for any function. What I don't get is, what does it actually achieve? what does it mean? Are there any real life examples where we use a Jacobian?
0
votes
0answers
19 views

What is the representation of Jacobian Determinant? [duplicate]

We know that if we want to do a change of variable on multiple integral, characteristic method on pde, change of variable on finding Probability DF (continuous distribution), etc Why we use Jacobian? ...
24
votes
4answers
13k views

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

What is an intuitive proof of multivariable changing of variables formula (jacobian) without using mapping and/or measure theory? I was thinking that textbooks over-complicate the proof. If possible,...
15
votes
4answers
2k views

Is advanced college math (eg analysis, abstract/linear algebra, topology) supposed to be as intuitive as elementary math? [closed]

So I don't know if I'm not smart enough for math, but lately, it seems to me as if some advanced topics are just too unintuitive in my opinion. For example, I have no idea what eigenvalues, ...
5
votes
4answers
553 views

Chain rule for partial derivatives intuition

Can somebody give me an intuitive explanation for the below equations. I'm not sure how they come about and how they can be perceived logically. $$\frac{\partial z}{\partial s} =\frac{\partial f}{\...
11
votes
1answer
899 views

A limit and a coordinate trigonometric transformation of the interior points of a square into the interior points of a triangle

The coordinate transformation (due to Beukers, Calabi and Kolk) $$x=\frac{\sin u}{\cos v}$$ $$y=\frac{\sin v}{\cos u}$$ transforms the square domain $0\lt x\lt 1$ and $0\lt y\lt 1$ into the ...
3
votes
1answer
3k views

Jacobian Matrix in dynamical systems

Can someone explain what exactly the Jacobian matrix is (specifically in its application to dynamical systems) and maybe give an example of how to compute it? It really confuses me...and I haven't ...
12
votes
1answer
521 views

What exactly does $\frac{\partial(y_1,\dots,y_m)}{\partial(x_1,\dots,x_n)}$ refer to?

I have been asking a rather few questions of this nature lately, maybe I'm starting to realise math notation isn't as uniform as I initially thought it would be... Question: Does this notation $$\...
0
votes
2answers
569 views

What is the geometrical interpretation of determinant of a matrix in general? [duplicate]

My question is simple (and maybe I am wrong asking this question even) what is the geometrical interpretation of determinant of a matrix in general ? I could not think anything.
2
votes
2answers
309 views

Why is the definition of derivative what it is?

In our lectures, we've been taught the following: We say that $f:\mathbb{R}^3\to\mathbb{R}$ is differentiable at a point $X$,iff there exists $\alpha\in\mathbb{R}^3$ such that $$\epsilon (H)=\frac{...
0
votes
3answers
552 views

Stable Matrices?

My question is quite simple: how can you determine the stability of a given dynamical system? For example, what values of a make the following matrix stable: \begin{bmatrix} 0&a \\ a&0 \end{...
0
votes
0answers
779 views

relationship of gamma and beta functions

I was reading a article on proving the relationship between gamma and beta functions but there are some things I don't understand. Then the variables are changed as Then they have obtained ...
1
vote
1answer
77 views

Chronicles of Discoveries

First I'd like to bring an example to make myself more clear. I know what the Jacobian matrix is and where, how and why it is used (some examples, at least) . But still I can't get it's geometrical ...
0
votes
2answers
60 views

Understanding polar change of variables for $\int_{-a}^adx\int_{-\sqrt{a^2-x^2}}^\sqrt{a^2-x^2}\frac{a}{\sqrt{a^2-x^2-y^2}}dy$

I have stumbled upon this expression in my textbook $$\frac{1}{2}S=\int_{-a}^{a}dx\int_{-\sqrt{a^2-x^2}}^{\sqrt{a^2-x^2}} \frac{a}{\sqrt{a^2-x^2-y^2}}dy$$ It stated that after passing to polars it ...

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