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.
7,091 questions
0 votes
1 answer
61 views
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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2 votes
0 answers
65 views
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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1 vote
0 answers
79 views
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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0 votes
0 answers
70 views
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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2 votes
0 answers
57 views
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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1 vote
0 answers
32 views
Transformation of partials in accelerating reference frame
I am working on a traveling wave problem and I am struggling to convince myself I've transformed to partial derivatives in a co-moving reference frame correctly. My previous post has a more background ...
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0 votes
0 answers
59 views
Vector Laplacian in Frenet–Serret coordinates with zero torsion
I am trying to follow this paper:https://accelconf.web.cern.ch/p03/papers/WPAB088.pdf and I would like to derive the Laplacian of a divergence-free vector field where the space curve has zero torsion ...
- 101
2 votes
1 answer
260 views
What is the need for the third assumption (premise) in the integral calculator?
\begin{equation} a \neq 20 \\ \implies a-20 \neq 0 \\ \implies (a-20)^2 \neq 0 \\ \implies (a-20)^2 > 0\\ \implies [(a-20)^2 + (b-20)^2] > 0 \\ \end{equation} \begin{equation} a \neq 20 \\ \...
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