# Tag Info

### Why does it seem the inner curls within a surface always cancels in order for greens theorem to be true

Imagine you know that the fundamental theorem of calculus: $$\int_a^b f'(x) dx = f(b) - f(a)$$ only holds for differentiable functions defined on $[0,1]$. Then how would you show, say, that for ...
• 2,502

### Torsion-free covariant derivative

$\nabla_XY$ is a vector field and hence acts as a derivation on $C^\infty(M)$, i.e. $\nabla_XY(fg)=f\nabla_XY(g)+g\nabla_XY(f)$ for any $f,g\in C^\infty(M)$. The object $X\circ Y$ is a second order ...
• 2,543

### "field" vs. "vector field"

No, the field in vector field comes from physics — electric field, gravitational field, magnetic field …. The word field in English has these two different mathematical meanings. In other languages, ...
• 118k
Accepted

• 118k
Accepted

• 2,255
Accepted

### Show that a frame field is (not) holonomic - only possible by going to the coframe?

$X$ and $Y$ are coordinate vector fields if and only if their Lie bracket $[X,Y]=0$. In your case, $[X,Y]=-\partial_y$.
• 118k

• 78.4k

### Concatenation of $f$-related vector fields/tangent vectors

So I found the problem. The concatenation encountered in the Lie bracket, is ,in fact, not defined as I did it in my question, but as follows. Let $h \in \mathcal{C}^\infty (M)$. Then [X^1,X^2]h: p ...
• 283
Accepted

• 22.5k
### Evaluating $\displaystyle\iint\limits_{A}\left(x^3-y^2z^4,2y^3,z^3-3y^2z\right)\cdot\overrightarrow{n}~\mathrm{d}S$, where $A$ is the unit sphere
The divergence of $F$ is equal to $3$ on the sphere, but not in the ball. The answer C is correct.