Questions tagged [jacobian]
In multivariable calculus, the jacobian matrix of a smooth map at a given point is the matrix of its partial derivatives evaluated at this point.
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The Gateaux derivative and the Jacobian
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Is it accurate to claim that the Gateaux derivative is the continuous linear operator represented by the Jacobian matrix? That is, the Jacobian is the finite dimensional equivalent of the ...
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The Jacobian and Integration of $n$-dimensional polar coordinates
This is my first proof and is most likely going to be crude. Please while reading comment tips about ways to improve my writing.
Polar coordinates are transformed through: $$x=r\cos(\theta) \qquad y=r\...
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Calculate $\iint_D|\cos(x+y)|dxdy,\quad D=[0,\pi]\times[-\pi,\pi]$
Calculate the double integral:
$$I=\iint_D|\cos(x+y)|\ dx\ dy,\quad D=[0,\pi]\times[-\pi,\pi]$$
Using Desmos I can see that the function forms four separate "leaves" on the rectangle $D$. ...
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Why is $H^1(G_k, \mathrm{Jac}_X) \cong H^1(G_k, \mathrm{Pic}(\overline{X}))$ in Milne’s Arithmetic Duality Theorems, Theorem A.7?
In Arithmetic Duality Theorems (2nd ed.), Chapter I, Theorem A.7, Milne considers a complete smooth curve $X$ over a quasi-finite field $k$ with function field $K = k(X),ドル and gives the following ...
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The meaning of the elements of the inverse of the Jacobian (and Hessian)
Page 147, equation (3.9.9, of "The Finite Element Method Linear Static and Dynamic Finite Element Analysis" by Thomas J. R. Hughes contains the following formula
$$
\begin{bmatrix}
\dfrac{\...
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Meaning of the discriminant of a general Jacobian
Let's say I have two functions: $y(a,b)$ and $z(a,b),ドル and I use the following Jacobian:
$J=\left(\begin{array}{cc}\frac{\partial y}{\partial a} & \frac{\partial y}{\partial b} \\ { \frac{\partial ...
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Let $u(x,y)$ and $v(x,y)$ be defined by $x=f(u,v)$ and $y=g(u,v),ドル find an expression for $\frac{\partial u}{\partial x}$
This is a question from a GRE Math Subject practice test. $u, v, f, g$ are all differentiable real-valued functions. The possible answers are
(A) $\frac{\partial f}{\partial u}$
(B) $\frac{\partial g}{...
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Calculate the Jacobian of a multivariable function, differentiating the euclidian norm
I got stuck on a problem from my calculus 3 workbook. One part of the problem is calculating the Jacobian matrix of a function $f:\mathbb{R}^n\times \mathbb{R}^n\to \mathbb{R}^{n}$ defined as:
$$f(x,y)...