Questions tagged [vector-analysis]
Questions related to understanding line integrals, vector fields, surface integrals, the theorems of Gauss, Green and Stokes. Some related tags are (multivariable-calculus) and (differential-geometry).
6,715 questions
0 votes
1 answer
61 views
Is using determinants like this for vector algebra standard?
It is known that, $$ \nabla \cdot (\mathbf{A} \times \mathbf{B}) = \mathbf{B} \cdot (\nabla \times \mathbf{A}) - \mathbf{A} \cdot (\nabla \times \mathbf{B}) $$ The straightforward way to prove this ...
- 11
0 votes
1 answer
177 views
Gradient Theorem Motivation via Generalized Stokes' and the Musical Isomorphisms
I attempted to motivate/derive the classical vector calculus Gradient Theorem $$\int_\gamma \vec{\nabla}f \cdot d\vec{r} = f(\vec{r}(b)) - f(\vec{r}(a))$$ with a non-conventional path of using the ...
- 105
1 vote
0 answers
78 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
0 answers
82 views
Is there a trick to finding the normal function of an ellipse given its parametric definition?
I’m trying to solve a calculus problem posed like this (N is the unit normal function and T is the unit tangent function): Use the formula $\textbf{N} = \frac{d\textbf{T}/dt}{|d\textbf{T}/dt|}$ to ...
- 1
6 votes
1 answer
148 views
Difference between Helmholtz-Leray decomposition and Helmholtz decomposition.
I'm trying to understand the Leray projection $\mathbb{P}$. Here is Wikipedia's definition: One can show that a given vector field $\mathbf{u}$ on $\mathbb {R} ^{3}$ can be decomposed as $$\mathbf{u}=...
- 6,872
0 votes
2 answers
120 views
I have trouble understanding the statement of kepler's second law. [closed]
Kepler's second law state: A planet moves in a plane, and the radius vector (from the sun to the planet) sweeps out equale area in equale time. To show that the force being central is equivalent to ...
1 vote
1 answer
114 views
Directional derivative of a vector field
Suppose $X: \mathbb{R}^3 \rightarrow \mathbb{R}^3$ is a vector field in $\mathbb{R}^3$, and let $$ Y = (X \cdot \nabla) X $$ be the directional derivative of $X$ in the direction of $X$. Then my ...
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6 votes
1 answer
299 views
A geometric proof of Hlawka inequality
Problem Statement: Let $\triangle ABC$ be a planar triangle with centroid $G$, and let $A_1$, $B_1$, $C_1$ be the midpoints of sides $BC$, $CA$, $AB$, respectively. For any point $P$ in the plane, ...
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