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Here's a bunch of problems, some of which I think I've done right, others I've attempted, others I have no clue. Any help would be appreciated, and please let me know if the one's I've done are right:
1. Let [itex]M_1 \subset \mathbb{R}^{n_1},\ M_2 \subset \mathbb{R}^{n_2[/itex] be oriented manifolds without boundary, of dimensions [itex]k_1[/itex] and [itex]k_2[/itex] respectively.
Show [itex]M_1 \times M_2[/itex] is a manifold without boundary in [itex]\mathbb{R}^{n_1 + n_2}[/itex] with a natural orientation induced by [itex]M_1[/itex] and [itex]M_2[/itex]
Well, [itex]\forall \, x_1 \in M_1[/itex], there is an open set [itex]U_{x_1}[/itex] containg [itex]x_1[/itex] and an open set [itex]V_{x_1} \subset \mathbb{R}^{n_1}[/itex], and a diffeomorphism [itex]h_{x_1} : U_{x_1} \to V_{x_1}[/itex] such that
[tex]h_{x_1}(U_{x_1} \cap M_1) = V_{x_1} \cap (\mathbb{R}^{k_1} \times \{ 0 \})[/tex], and something similar for points in [itex]M_2[/itex]. Let [itex]x = (x_1,\, x_2)[/itex] be any point in [itex]M_1 \times M_2[/itex]. Define a function [itex]h_x : U \to V[/itex] where [itex]U = U_{x_1} \times U_{x_2}[/itex] by:
[tex]h_x(u_1, u_2) = (h_{x_1}(u_1), h_{x_2}(u_2))[/tex]
Then define the permutation p by:
[tex]p(y_1, \dots , y_{n_1 + n_2}) = (y_1,\dots ,y_{k_1}, y_{n_1 + 1} ,\dots, y_{n_1 + k_2}, y_{k_1 + 1}, \dots , y_{n_1}, y_{n_1 + k_2 + 1}, \dots , y_{n_1 + n_2})[/tex]
Then define [itex]H_x = p \circ h_x[/itex] for each x. This function satisifies the conditions required to make [itex]M_1 \times M_2[/itex] a manifold. (Is it right?)
2. Let S be the set defined by the equations:
[tex]x^2 + y^2 + z^4 = 3,\ x^3 - y^3 + z(1 + xy) = 2[/tex]
Let [itex]f(x, y, z) = e^{x + yz} + x^3y[/itex]
Show that, for P = (1, 1, 1) and some [itex]\epsilon[/itex] > 0, [itex]M = S \cap \mathcal{B}_P (\epsilon )[/itex] is a manifold, where [itex]\mathcal{B}_P (\epsilon )[/itex] is the open ball of radius [itex]\epsilon[/itex] centered at point P.
Don't really know how to do this one.
1. Let [itex]M_1 \subset \mathbb{R}^{n_1},\ M_2 \subset \mathbb{R}^{n_2[/itex] be oriented manifolds without boundary, of dimensions [itex]k_1[/itex] and [itex]k_2[/itex] respectively.
Show [itex]M_1 \times M_2[/itex] is a manifold without boundary in [itex]\mathbb{R}^{n_1 + n_2}[/itex] with a natural orientation induced by [itex]M_1[/itex] and [itex]M_2[/itex]
Well, [itex]\forall \, x_1 \in M_1[/itex], there is an open set [itex]U_{x_1}[/itex] containg [itex]x_1[/itex] and an open set [itex]V_{x_1} \subset \mathbb{R}^{n_1}[/itex], and a diffeomorphism [itex]h_{x_1} : U_{x_1} \to V_{x_1}[/itex] such that
[tex]h_{x_1}(U_{x_1} \cap M_1) = V_{x_1} \cap (\mathbb{R}^{k_1} \times \{ 0 \})[/tex], and something similar for points in [itex]M_2[/itex]. Let [itex]x = (x_1,\, x_2)[/itex] be any point in [itex]M_1 \times M_2[/itex]. Define a function [itex]h_x : U \to V[/itex] where [itex]U = U_{x_1} \times U_{x_2}[/itex] by:
[tex]h_x(u_1, u_2) = (h_{x_1}(u_1), h_{x_2}(u_2))[/tex]
Then define the permutation p by:
[tex]p(y_1, \dots , y_{n_1 + n_2}) = (y_1,\dots ,y_{k_1}, y_{n_1 + 1} ,\dots, y_{n_1 + k_2}, y_{k_1 + 1}, \dots , y_{n_1}, y_{n_1 + k_2 + 1}, \dots , y_{n_1 + n_2})[/tex]
Then define [itex]H_x = p \circ h_x[/itex] for each x. This function satisifies the conditions required to make [itex]M_1 \times M_2[/itex] a manifold. (Is it right?)
2. Let S be the set defined by the equations:
[tex]x^2 + y^2 + z^4 = 3,\ x^3 - y^3 + z(1 + xy) = 2[/tex]
Let [itex]f(x, y, z) = e^{x + yz} + x^3y[/itex]
Show that, for P = (1, 1, 1) and some [itex]\epsilon[/itex] > 0, [itex]M = S \cap \mathcal{B}_P (\epsilon )[/itex] is a manifold, where [itex]\mathcal{B}_P (\epsilon )[/itex] is the open ball of radius [itex]\epsilon[/itex] centered at point P.
Don't really know how to do this one.
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