# What is Jacobian: Definition and 168 Discussions

In mathematics, a Jacobian, named for Carl Gustav Jacob Jacobi, may refer to:

Jacobian matrix and determinant
Jacobian elliptic functions
Jacobian variety
Intermediate Jacobian

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1. ### Classification of Equlibrium Points

I hope this is more properly laid out? We previously established that the stationery points were (1,1) and (-1,1) For this first stage I now need to create the elements of a Jacobian maitrix using partial differentation. I am confused by reference to the chain rule. Am I correct that for dx/dt...
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### Jacobian: how to change limits of integration?

Hello, I have to compute a double integral of the form ## \int_{0}^{\infty} \int_{0}^{\infty} f(u,v) du dv##, where ##f(u,v)## is not relevant. The following change of variable is advised as a hint: ## u = zt ## and ## v = z(1-t)##. From there, I can reformulate with respect to ##z## and...
3. ### I Confusion about special case of Jacobian

I am used to the usual definition of the Jacobian (when the talk is about derivatives) as the Jacobian matrix for multi-valued functions. However, in the 1995 edition of the introductory book "Basic Training in Mathematics: A fitness program for science students" on page 45 , equations 2.2.22...
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### Change of variable for Jacobian: is there a method?

Hello, This problem comes just prior to introducing change of variables with Jacobian. Given the following region in the x-y plane, I have to choose (with justification) the correct change of variables associated, for ##u\in [0,2]## and ##v \in [0,1]##. The correct choice here is a), but I do...
5. ### When to use the Jacobian in spherical coordinates?

Greetings! here is the solution which I undertand very well: my question is: if we go the spherical coordinates shouldn't we use the jacobian r^2*sinv? thank you!

12. ### I Multiple integral Jacobian confusion

Consider a continuous charge distribution in volume ##V'##. Draw a closed surface ##S## inside the volume ##V'##. ___________________________________________________________________________ Consider the following multiple integral: ##\displaystyle B= \iint_S \Biggl( \iiint_{V'}...
13. ### I What if the Jacobian doesn't exist at finite points in domain of integral?

Consider a one to one transformation of a ##3##-##D## volume from variable ##(x,y,z)## to ##(t,u,v)##: ##\iiint_V dx\ dy\ dz=\int_{v_1}^{v_2}\int_{u_1}^{u_2}\int_{t_1}^{t_2} \dfrac{\partial(x,y,z)}{\partial(t,u,v)} dt\ du\ dv## ##(1)## Now for a particular three dimensional volume, is it...
14. ### I Time evolution of a Jacobian determinant

In this paper ##J=\frac{\partial f_1(X_1)}{\partial X_1}\frac{\partial f_2(X_2)}{\partial X_2}\frac{\partial f_3(X_3)}{\partial X_3}## where ##f_2(X_2),f_1(X_1),f_3(X_3)## evolves with time. Now using this ##\dot J=\frac{d}{dt}(\frac{\partial f_1(X_1)}{\partial X_1}\frac{\partial...
15. ### I Calculating Jacobian Determinant

I came across a line in this paper at page (2) at right side 2nd para where it is written ##d^3x=Jd^3X## where ##J## is the Jacobian and x and X are the positions of the fluid elements at time ##t_0## and ##t## respectively. Here what I have concluded that ##x_i=f(X_i)## where the functional...
16. ### MHB The Jacobian and area differential

I don't understand the following definition. If we let $u=\langle u,v \rangle$ , $p=\langle p,q\rangle,$ $x=\langle x,y \rangle$,then (x,y)=T(u,v) is given in vector notation by x=T(u). A coordinate transformation T(u) is differentiable at a point p , if there exists a matrix J(p) for which...
17. ### Problem solving this volume using Jacobi's Determinant

Homework Statement Find the value of the solid's volume given by the ecuation 3x+4y+2z=10 as ceiling,and the cilindric surfaces 2x^2=y x^2=3*y 4y^2=x y^2=3x and the xy plane as floor.The Attempt at a Solution I know that we have to give the ecuation this form: ∫∫z(x,y)dxdy= Volume So, in fact...
18. ### A Hessian as "Square" of Jacobian?

Hi, Is there a way of representing the Laplacian ( Say for 2 variables, to start simple) ##\partial^2(f):= f_{xx}+f_{yy} ## as a "square of Jacobians" ( More precisely, as ##JJ^T ; J^T ## is the transpose of J, for dimension reasons)? I am ultimately trying to use this to show that the...
19. ### I Property of Jacobian Determinant

We can denote the jacobian of a vector map ##\pmb{g}(\pmb{x})## by ##\nabla \pmb{g}##, and we can denote its determinant by ##D \pmb{g}##. We were asked to prove that ##\sum_j \frac{\partial ~ {cof}(D \pmb{g})_{ij}}{\partial x_j} = 0## generally holds so long as the ##g_i## are suitably...
20. ### MHB Jacobian of the transformation T:x=u, y=uv

how do you graph this in Desmos ? Assume the rest of the calculation is correct much thank you ahead...:cool:
21. ### How to know which variable comes first in the Jacobian?

Homework Statement Find the Jacobian of the transformation: x = e^{-r}sinθ , y = e^rcosθ Homework EquationsThe Attempt at a Solution formula for Jacobian is absolute value of the determinant \begin{vmatrix} \frac {∂x}{∂u} & \frac {∂x}{∂v}\\ \frac {∂y}{∂u} & \frac {∂y}{∂v}\\ \end{vmatrix}...
22. ### Jacobian of a Lorentz transformation

Homework Statement I've never encountered Jacobians before, and having read up on them a bit I find the wording of the last part of this question confusing: A set of coordinates ##x'_{\mu}## in frame B is obtained from the set ##x_{\mu}## in frame A, by boosting B w.r.t A with speed beta along...
23. ### Jacobian matrix and Navier Stokes equation

Hi, in first attachment/picture you can see the generalized navier stokes equation in general form. In order to linearize these equation we use Beam Warming method and for the linearization process we deploy JACOBİAN MATRİX as in the second attachment/picture. But on my own I can ONLY obtain the...
24. ### Jacobian of a coordinate system wrt another system

Homework Statement Homework EquationsThe Attempt at a Solution Jacobian of the coordinate- system (## u_1, u_2##) with respect to another coordinate- system (x,y ) is given by J = ## \begin{vmatrix} \frac { \partial {u_1 } } {\partial {x } } & \frac { \partial {u_1 } } {\partial {y} } \\...
25. ### I Coherent operations on Jacobian matrices

Is there a notion of “coherent” operations on Jacobian matrices? By this I mean, an operation on a Jacobian matrix A that yields a new matrix A' that is itself a Jacobian matrix of some (other) system of functions. You can ascertain whether A' is coherent by integrating its partials of one...
26. ### Load Flow Order of Jacobian Matrix Power System

I'm studying Newton Raphson Method in Load Flow Studies. Book has defined Jacobian Matrix and it's order as: N + Np - 1 N = Total Number of Buses Np = Number of P-Q Buses But in solved example they've used some other formula. I'm not sure if it's right. Shouldn't order be: N + Np - 1 N = 40 Np...
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### Finding the volume surrounded by a curve using polar coordinate

Homework Statement I tried to answer the following questions is about the curve surface z= f (x, y) = x^2 + y^2 in the xyz space. And the three questions related to each otherA.) Find the tangent plane equation at the point (a, b, a^2+ b^2) in curved surface z . The equation of the...

50. ### Kernel vector of statics Jacobian

Hi all, I was reading an article that utilized a 3x4 statics Jacobian and said to calculate the kernel vector: You can row by row, where Where Ai is the statics Jacobian with the ith column removed. The problem is I have a 3x3 statics Jacobian, so if I remove the ith column I will end up...