I have difficulty understanding the following Theorem
If U is open in ℝ^2, F: U \rightarrow ℝ is a differentiable function with Lipschitz derivative, and X_c=\{x\in U|F(x)=c\}, then X_c is a smooth curve if [\operatorname{D}F(\textbf{a})] is onto for \textbf{a}\in X_c; i.e., if \big[...
I read the book again and found out it's just the notation they use:
for any differentiable function f defined about x and any tangent vector \xi they set \xi(f)=D_\varphi(f) where \varphi \in \xi (they define a tangent vector as an equivalence class), so D_{\varphi_x}f=X(x)f
@quasar987: The...
Let \varphi be a one-parameter group on a manifold M, and let f be a differentiable function on M, the derivative of f with respect to \varphi is the defined as the limit:
\lim_{t\to 0} \frac{\varphi^*_t[f]-f}{t}(x)=\lim_{t\to 0}\frac{f\circ \varphi_x(t)-f\circ...
If I choose the chart (U_1,\phi_1), where U_1 is the lower semi-sphere, then the first condition is not met because the north pole is in the upper semi-sphere; also, since the points converge to the north pole, how do you find an integer N such that x_k \in U_1 for k>N?
I have a question regarding the following definition of convergence on manifold:
Let M be a manifold with atlas A. A sequence of points \{x_i \in M\} converges to x\in M if
there exists a chart (U_i,\phi_i) with an integer N such that x\in U_i and for all k>N,x_i\in U_i
\phi_i(x_k)_{k>N}...
According to duality principle, a bilinear function \theta:V\times V \rightarrow R is equivalent to a linear mapping from V to its dual space V*, which can in turn be represented as a matrix T such that T(i,j)=\theta(\alpha_i,\alpha_j). And this matrix T is diagonalizable, i.e...
There are line integral with respect to arc length and line integral with respect x/y.
I know \int_C Pdx+Qdy is useful to calculate the work. When do we need the line integral with respect to arc length?
Is there a name for the linear mapping f^*:\wedge^k(R^{m}_{f(p)}) \rightarrow \wedge^k(R^{n}_{p}) where f is a differentiable mapping from R^n \rightarrow R^m.
When k is 1, f* is called the adjoint of f. But what about k > 1?
Also can someone show me a proof of f^*(d\omega)=d(f^*\omega)...
I don't understand how order 1 alternating tensors fit the definition of alternating tensors. I think the order has to be at least 2 for the definition to make sense because only then can we talk about permutation.
So I think there's no difference between order 1 alternating tensors and...
A non-zero alternating tensor w splits the bases of V into two disjoint groups, those with \omega(v_1,\cdots,v_n)>0 and those for which \omega(v_1,\cdots,v_n)<0.
So when we speak of the orientation of a vector space, we need to say the orientation with respect to a certain tensor, correct?
To prove the rationals are dense in the reals, you need to prove for any real number, there exists a rational number which is arbitrarily close to the real number. I don't see this in your proof.
I'm wondering why 1/k! is needed in Alt(T), which is defined as:
\frac{1}{k!}\sum_{\sigma \in S_k} \mbox{sgn}\sigma T(v_{\sigma(1)},\cdots,v_{\sigma(k)})
After removing 1/k!, the new \mbox{Alt}, \overline{\mbox{Alt}}, still satisfies...