Hello,
I have managed to calculate that
\phi_{-t}^*\left( \displaystyle\frac{\partial}{\partial x_j} \Bigg|_{\phi_t(p)} \right)= \displaystyle\sum_{i,j} \frac{\partial \phi^i}{\partial x_j}(-t, \phi_t(p)) \frac{\partial}{\partial x_j} \Bigg|_p ,
although in Loring Tu's book the same...
Thanks for your response. I understand the 2nd concern but I still have trouble with the first. I cannot see why that statement is true. I know that \phi_t ^* Y is a vector field, so \phi_t^* Y |_p \in T_p M , i.e.
\phi_t^* Y |_p = \displaystyle\sum_i a_i^t (p) \frac{\partial}{\partial x_i...
Hello all I had a simple question that I am intuitively sure I know the answer to but can't quite prove it.
Suppose k is a polynomial in x and y, and k(x-1) = q for q some polynomial in y. Then is k = 0 ?
How do I verify that k must be equal to 0? I can see that to just get a polynomial...
Hello all, I was hoping someone would be able to clarify this issue I am having with the Lie Derivative of a vector field.
We define the lie derivative of a vector field Y with respect to a vector field X to be
L_X Y :=\operatorname{\frac{d}{dt}} |_{t=0} (\phi_t^*Y), where \phi_t is the...
Hello everyone, I just had a quick question I was hoping somebody could answer.
If f: M \times N \rightarrow P is a smooth map, where M,N and P are smooth manifolds, then is it true for fixed m that f_m : N \rightarrow P is smooth, where f_m (n) = f(m,n)?
Any help would be appreciated.
Hello everyone, I was wondering if I could get some advice for the following problem:
I have two algebraic sets X, X', i.e. X = V(J), Y = V(J'), and let I(X),I(Y) be the ideals of these sets, i.e. I(X) ={x \in X | f(x) = 0 for all x \in X}. I am trying to show that I(X \cap Y) is not always...