Questions tagged [partial-derivative]

For questions regarding partial derivatives. The partial derivative of a function of several variables is the derivative of the function with respect to one of those variables, with all others held constant.

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Is it possible to let $\frac{\partial f(x,y)}{\partial (-y)}=\frac{\partial f(x,a-y)}{\partial (a-y)}$ hold, where $a$ is a constant? [closed]

Let $f(x,a-y)$ be a function that is continuously differentiable in $y$, and $a$ be a constant. Under what conditions does the following equality hold? \frac{\partial f(x,y)}{\partial ...
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Chain rule when mixed partial multiplied by first order partial

Given $$r = \frac{{\partial s}}{{\partial t}}$$ If we take the partial derivative of $r$ with respect to $u$ $\frac{{\partial r}}{{\partial u}}$ and multiply it by $\frac{{\partial u}}{{\partial v}}$...
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Changing coordinates of $2$nd order partial operators

Let's work in $\mathbf R^n$. If we want to change coordinates $\mathbf x\to\mathbf r$, with them related like $$\mathbf x=\mathbf x(\mathbf r)$$ Then, the second order generic partial operator in ...
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Derivative of the determinant of a matrix with respect to the components

Given a square matrix $A$ $$A = \begin{bmatrix} a_{11} & \cdots & a_{1n} \\ \vdots & \ddots &\vdots\\ a_{n1} & \cdots & a_{nn} \end{bmatrix}$$ what is the ...
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Gradients of vertices on a grid

Say we have an irregular 2D grid, it can be viewed as a group of triangles composed, and we know the value of function v(x,y) on the plane at each vertex on the grid (The picture is just for ...
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Can the partial derivative of a multivariable function be such that it does not depend on the other variable? [duplicate]

I wanted to know if there is a multivariable function out there which when partially differentiated with respect to x returns a function only in x.(Not dependent on y(the other variable)). For example-...
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Help with multivariate function derivative algebra... Middle terms don't match as expected

Intro I want to compute $\frac{\partial^2 f}{\partial m^2}$ where we define $f(m,\sigma) = g(m, I(m, t))$ I asked about this in a prior question, however I'm realizing that I don't actually ...
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Derivation of equation from Euler's equation for a fluid

I'm having trouble with the formula for particle velocity of a standing wave in a tube, starting from conservation of mass and Euler's equation for a fluid. I've checked a few books and unfortunately ...
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Information coefficient as loss function of XGBoost

$$IC = \frac{\frac{1}{n}\hat{y}^Ty-\mathrm{E}\left[ \hat{y} \right] \mathrm{E}\left[ y \right]}{\sigma \left[ \hat{y} \right] \sigma \left[ y \right]}$$ XGBoost requires a gradient and a Hessian of ...
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Schwartz theorem

Can i relax the conditions on f of the Schwartz theorem in real analysis? That is: if $f: \Omega \to R^2$, s.t $f \in C^2(\Omega)$, then i can change the order of derivatives and the result is the ...
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Chain rule in differential of Gauss map

This is probably more of a calculus question than a geometry question. Let $N: S \rightarrow \mathbb{S}^2$ be the Gauss map. And let $\varphi : \mathbb{R}^2 \supseteq U \rightarrow S$ a ...
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Prove that in the canonical basis $\mathrm{grad}\,f = \sum_{j=1}^{n} \frac{\partial f}{\partial x_j}\,e_i$.

Reading Do Carmo I found this exercise: Given a differentiable function $f:\mathbb{R}^n \to \mathbb{R}$ the vector field (gradient) is defined by $$\langle \mathrm{grad}\,f(p) \rangle =df_p(u)$$ ...
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