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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Prove that the Euclidean norm is not differentiable at the origin
Let the scalar field $f : {\Bbb R}^2 \to {\Bbb R}$ be defined by $f(x,y) := \sqrt{x^2 + y^2}$. Prove that the partial derivatives of $f$ at $(0,0)$ do not exist and, thus, the gradient $\nabla f (0,0)$...
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Finding partial derivatives with direct and indirect dependencies
A student I am tutoring in multivariable calculus was given the following problem:
Suppose z = f(x, y), and we have an equation F(x, y, z) = k for some function F and
constant k. Use the chain rule to ...
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Why do we supposed to use Log function in Logistic regression's cost calculation. [closed]
I went through the Logistic regression function, which involves linear functions along with sigmoid technique. While calculating the cost, In Logistic regression they are using cross-entropy loss ...
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Applying Leibniz's Rule to Double Integrals with Variable Limits
Consider the following double integrals:
$$G_1(z_1, z_2) = \int^{z_1}_{0} \int_{0}^{z_2 + \frac{\alpha_1}{\alpha_2}(z_1 - x_1)} \varphi(x_1, x_2) \, dx_2 \, dx_1$$
$$G_2(z_1, z_2) = \int^{z_2}_{0} \...
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Suppose $D^3f$ exists, prove $D_AD_AD_A f$ exists.
Let $A,B,C$ be (banach) normed spaces. On $A\times B$ we consider the supremum norm. Let $X\subseteq A \times B$ be an open set.
Let $f: X \rightarrow C$ and suppose its third differential map exists,...
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Confusion in PDE multivariate chain rule
I have a doubt regarding the multivariate chain rule PDE.
Consider an arbitrary function $\phi(x+y+z,x^2+y^2-z^2)=0$. We have to eliminate the function & form a PDE.
The solution as follows:
Let $...
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Lie bracket and independence on the coordinates
I am studying differentiable topology and I am facing the definition on vector field and Lie bracket. If $M$ is an $m-$manifold and $V,W:T\to TM$ are vector fields on $M$, we define the Lie bracket of ...
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Covariant derivatives of tensor densities
The answer to this question proves the following:
\begin{equation}
\partial_\sigma \sqrt{-\det{\mathrm{g}}} = \frac12 \sqrt{-\det{\mathrm{g}}} \;g^{\alpha\beta}\partial_\sigma g_{\alpha\beta} \qquad (...
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Derivative of a family of functions with respect to the parameter of the family
I always struggle when working with partial derivatives in the context of differential geometry. When I read computations done by some other person, I try to see at each step what is the domain and ...
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derivative with vectors
Given $\mathbf X(s_1, s_2, v) = \Delta t\mathbf v+\sigma s_1(\hat{\mathbf n}_1+\mathbf v)+\tau s_2(\hat{\mathbf n}_2+\mathbf v)$, is it possible to express $\hat{\mathbf n}_1\cdot\nabla_{\mathbf X}$ ...
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If an operator is invariant with respect to 2D rotation, is it also invariant with respect to 3D rotation?
Problem: For a function $f(x,y,z)$ and a rotational change of coordinates $(x,y,z)\to (u,v,w)$, the following relation holds
$$\frac{\partial^2 f}{\partial x^2}+\frac{\partial^2 f}{\partial y^2}+\frac{...
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Partial and total derivative on multivariable functions
This is a weird question.
I've encountered this problem on Lagrangians so much that I have started doubting. A similar question is at:
Partial Derivative vs Total Derivative: Function depending ...
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What is the derivative of the function $f(x) = |x| \sin(\frac{1}{x})$ at $x = 0$?
My Question:
What is the derivative of the function $f(x) = |x| \sin\left(\frac{1}{x}\right)$ at $x = 0$?
Background:
This problem arises when studying the differentiability of functions involving ...
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Total derivative for mixed higher order
I understand total derivative as a best linear approximation. Is there something similar for higher order derivatives where the orders are different for each variable?
Let me give an example. Assume $...
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Understanding the mathematical expression of the operator of an infinitesimal isotropic expansion
While reading the paper: Chudnovsky, A. "Crack layer theory" No. NASA-CR-174634 1984, I came across the following expression for an operator of an infinitesimal isotropic expansion:
$\delta^{...
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