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.
7,092 questions
0
votes
1
answer
42
views
Applying Leibniz's Rule to Double Integrals with Variable Limits
Consider the following double integrals:
$$G_1(z_1, z_2) = \int^{z_1}_{0} \int_{0}^{z_2 + \frac{\alpha_1}{\alpha_2}(z_1 - x_1)} \varphi(x_1, x_2) \, dx_2 \, dx_1$$
$$G_2(z_1, z_2) = \int^{z_2}_{0} \...
- 3
1
vote
0
answers
39
views
Suppose $D^3f$ exists, prove $D_AD_AD_A f$ exists.
Let $A,B,C$ be (banach) normed spaces. On $A\times B$ we consider the supremum norm. Let $X\subseteq A \times B$ be an open set.
Let $f: X \rightarrow C$ and suppose its third differential map exists,...
- 173
0
votes
1
answer
66
views
Confusion in PDE multivariate chain rule
I have a doubt regarding the multivariate chain rule PDE.
Consider an arbitrary function $\phi(x+y+z,x^2+y^2-z^2)=0$. We have to eliminate the function & form a PDE.
The solution as follows:
Let $...
- 3
7
votes
2
answers
379
views
Lie bracket and independence on the coordinates
I am studying differentiable topology and I am facing the definition on vector field and Lie bracket. If $M$ is an $m-$manifold and $V,W:T\to TM$ are vector fields on $M$, we define the Lie bracket of ...
- 877
2
votes
0
answers
68
views
Covariant derivatives of tensor densities
The answer to this question proves the following:
\begin{equation}
\partial_\sigma \sqrt{-\det{\mathrm{g}}} = \frac12 \sqrt{-\det{\mathrm{g}}} \;g^{\alpha\beta}\partial_\sigma g_{\alpha\beta} \qquad (...
- 21
2
votes
1
answer
59
views
Derivative of a family of functions with respect to the parameter of the family
I always struggle when working with partial derivatives in the context of differential geometry. When I read computations done by some other person, I try to see at each step what is the domain and ...
1
vote
0
answers
79
views
derivative with vectors
Given $\mathbf X(s_1, s_2, v) = \Delta t\mathbf v+\sigma s_1(\hat{\mathbf n}_1+\mathbf v)+\tau s_2(\hat{\mathbf n}_2+\mathbf v)$, is it possible to express $\hat{\mathbf n}_1\cdot\nabla_{\mathbf X}$ ...
- 11
0
votes
2
answers
54
views
If an operator is invariant with respect to 2D rotation, is it also invariant with respect to 3D rotation?
Problem: For a function $f(x,y,z)$ and a rotational change of coordinates $(x,y,z)\to (u,v,w)$, the following relation holds
$$\frac{\partial^2 f}{\partial x^2}+\frac{\partial^2 f}{\partial y^2}+\frac{...
- 989
0
votes
0
answers
71
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 ...
- 99
1
vote
0
answers
136
views
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 ...
- 35
2
votes
0
answers
60
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 $...
- 21
0
votes
0
answers
39
views
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^{...
- 71
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 ...
- 71
1
vote
1
answer
188
views
Why the second derivative test use $f_{xx}f_{yy}-f^2_{xy}$ rather $D^2_uf$ to test?
The second-order derivative determine whether this point is a critical point.
$y=f(x,y)$, and $\vec{u}=(a,b)$ is a unit vector.
$$
D_{u}f=\nabla{f}\cdot\vec{u}=f_xa+f_yb$$
$$
D_{u}^{2}f=f_{xx}a^{2}+...
- 81
0
votes
0
answers
60
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