Jacobian Definition and 163 Threads

Proving this geometrically [1] gives ##J = r.## Why is the ##-r## one wrong? Why is ##(x, y) \rightarrow (\theta, r)## is different from ##(x, y) \rightarrow (r, \theta)##? Edit: In Paul's Notes [2] it seems like ##J## is always positive, but online says it can be negative... [1] The first...

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...

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...

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...

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...

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!

The way I approach it was, we're looking for det(H) where H = h(u, v) $$H = \begin{bmatrix} du/da & du/db \\ dv/da & dv/db \end{bmatrix} * \begin{bmatrix} da/dx & da/dy \\ db/dx & db/dy \end{bmatrix}$$ I just multiply those two matrices and then get the determinant. The answer is $$16((ln x)^2...

While learning about Special Relativity I learned that we use the Transformation matrix to alter the space .This matrix differs for Contravariant and Covariant vectors.Why does it happen?,Why one kind of matrix (Jacobian) for basis vectors and other kind(Inverse Jacobian) for gradient...

I'm used to calculating Jacobians with several functions, so my only question would be how do I approach solving this one with only one function but three variables? I think our function becomes (s^2+sin(rt)-3)/since we are looking for J(f/s). So then would our Jacobian simply be J=[∂f/∂s...

I got that ##{x_u}{y_v}-{x_y}{y_u}=####\frac{1}{\frac{1}{{x_u}{y_v}}-\frac{1}{{y_u}{x_v}}}##. But this implies that ##{x_u}{x_v}{y_u}{y_v}=-1## and I don't see how that is true?

From the equations, I can find Jacobians: $$J = \frac {1}{4(x^2 + y^2)} $$ But, I confuse with the limit of integration. How can I change it to u,v variables? Thanks...

On Page 406 of Nolting Theoretical Physics 1 he has the following notation for the Jacobian determinant $$\frac{\partial( x_{1}, x_{2})}{\partial (y_{1}, y_{2})} = \begin{vmatrix} \left (\frac{\partial x_{1}}{\partial y_{1}} \right )_{y_{2}}& \left ( \frac{\partial x_{1}}{\partial y_{2}}...

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'}...

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...

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...

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...

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...

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...

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...

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...

how do you graph this in Desmos ? Assume the rest of the calculation is correct much thank you ahead...:cool:

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}...

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...

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...

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} } \\...

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...

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...

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...

I need help in understanding how Jacobian Elliptic Functions are interpreted as inverses of Elliptic Functions. Please reference the wiki page on Jacobian Elliptic functions: https://en.wikipedia.org/wiki/Jacobi_elliptic_functions For example, if $$u=u(φ,m)$$ is defined as $$u(φ,m) =...

Consider an algebraic variety, X which is a smooth algebraic manifold specified as the zero set of a known polynomial. I would appreciate resource recommendations preferably or an outline of approaches as to how one can compute the period matrix of X, or more precisely, of the Jacobian variety...

Let the matrix of partial derivatives ##\displaystyle{\frac{\partial y^{j}}{\partial y^{i}}}## be a ##p \times p## matrix, but let the rank of this matrix be less than ##p##. Does this mean that some given element of this matrix, say ##\displaystyle{\frac{\partial y^{1}}{\partial u^{2}}}##, can...

hi, I always see that jacobian matrix is derived for just 2 dimension ( ıt means 2x2 jacobian matrix) in books while ensuring the coordinate transformation. After that kind of derivation, books say that you can use same principle for higher dimensions. But, I really wonder if there is a proof...

Let $U$ be a compact set in $\mathbb{R}^k$ and let $f:U\to\mathbb{R}^n$ be a $C^1$ bijection. Consider the manifold $M=f(U)$. Its volume distortion is defined as $G=det(DftDf).$ If $n=1$, one can deduce that $G=1+|\nabla f|^2$. What happens for $n>1$? Can one bound from below this $G$? If...

Hi, I have to resample images taken from camera, whose target is a spherical object, onto a regular grid of 2 spherical coordinates: the polar and azimutal angles (θ, Φ). For best accuracy, I need to be aware of, and visualise, the "footprints" of the small angle differences onto the original...

Ok, I've got these functions to get the x (right), y (up) and z (forward) coordinates to plot with my computer program: x = r*Math.cos(a)*Math.sin(o) y = r*Math.sin(a) z = -r*Math.cos(a)*Math.cos(o) It's the equations of a sphere where I've placed the origin (o,a,r) = (0,0,0) of the source...

Hi For a sphere: x = r*cos(a)*sin(o) y = r*sin(a) z = -r*cos(a)*cos(o) where r is radius, a is latitude and o is longitude, the directional derivative (dx,dy,dz) is the jacobian multiplied by a unit vector (vx,vy,vz), right? So i get: dx = cos(a)*sin(o)*vx - r*sin(a)*sin(o)*vy +...

Can someone help ... the following is supposed to be a Jacobian matrix ...= $$ \begin{pmatrix} \frac{ \partial F_1}{ \partial x_1} & \frac{ \partial F_1}{ \partial x_2} \\ \frac{ \partial F_2}{ \partial x_1} & \frac{ \partial F_2}{ \partial x_2} \\ \frac{ \partial F_3}{ \partial...

Apologies for perhaps a very trivial question, but I'm slightly doubting my understanding of Jacobians after explaining the concept of coordinate transformations to a colleague. Basically, as I understand it, the Jacobian (intuitively) describes how surface (or volume) elements change under a...

Hey guys, Im studying for an exam and don't fully understand the jacobian process. Speciffically how you can differential the middle colum with respect to theta_3. Please view attached. So from step 1 to 2. Thanks.

Homework Statement I'm trying to write a program to solve a system of 3 non-linear equations using the Newton-Raphson method. I'm stuck on trying to figure out what the formula for a system of 3 unknowns is. I can't remember the derivation at all and after endless hours of googling and looking...

My maths teacher taught me a shortcut for finding area bounded by curves of the form: $$|as+by+c|+|Ax+By+C|=d$$ Shortcut: Let required area be ##A_0## and new area after "transformation" be ##A## Then, $$A_0\begin{vmatrix} a& b\\ A& B\end{vmatrix}=A=2d^2$$ All I understood was the ##A=2d^2##...

I don't understand why the ##W' \rightarrow \tau \nu## doesn't show a Jacobian peak whereas the ##W' \rightarrow (e/\mu) \nu## decay modes do...?? Is it because the ##\tau## decays even further (before measured) and gives additional Missing Transverse Energy? Is it the same for W \rightarrow...

## \int_{0}^{∞}\int_{0}^{∞} \frac{x^2+y^2}{1+(x^2-y^2)^2} e^{-2xy} dxdy ## ##u= x^2-y^2## ##v=2xy##I tried to find the jacobian and the area elements, I found it to be ## dA = \frac{1}{v} du dv ## I'm having problem finding the limits of u & v and getting rid of ##x^{2}+y^{2}##.

Homework Statement I will just post an image of the problem and here's the link if the above is too small: http://i.imgur.com/JB6FEog.png?1Homework EquationsThe Attempt at a Solution I've been playing with it, but I can't figure out a good way to "grip" this problem. I can see some things...

Why is it so that I can write: ##x'_i=A_{ij}x_j## where ##A_{ij}=\frac{\partial x'_i}{\partial x_j}##? Yes if the first expression is assumed it is clear to me why the coefficients have to be the partial derivatives, but why can we assume that we can always write it in a linear fashion in the...

Hi all, my question is rather a simple one and regards conformal transformations. On "Applied CFT" by P.Ginsparg, http://arxiv.org/pdf/hep-th/9108028.pdf , on page 10, gives the transformation rule of a quasi primary field and relates the exponent of 1.12 to the one of 1.10. My first question...

Homework Statement differentiate the function F(x,y) = f( g(x)k(y) ; g(x)+h(y) ) Homework Equations Standard rules for partial differentiation The Attempt at a Solution The Jacobian will have two columns because of the variables x and y. But what then? f is a multivariate function inside...

Homework Statement Don't understand why the inverse jacobian has the form that it does. Homework Equations $$ J = \begin{pmatrix} \frac{\partial{x}}{\partial{u}} & \frac{\partial{y}}{\partial{u}} \\ \frac{\partial{x}}{\partial{v}} & \frac{\partial{y}}{\partial{v}} \end{pmatrix} $$ $$...

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...

Hi, I'm looking at the Jacobian condition which is ## J= a \frac{dy_{0}}{ds}-b\frac{dx_0}{ds}## where the pde takes the form ##c= a\frac{\partial u}{\partial x} + b \frac{\partial u}{\partial y} ##, where ##a=\frac{\partial x}{\partial \tau } ##, ##b=\frac{\partial y}{\partial \tau }##...