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a ptwise orientation X_i is continuous iff locally w(X) has constant sign for some n-form w.
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If X_i is a continuous pointwise orientation then on a nghood U of p it has continuous representative Y_i. As w(X_1, …, X_n) = det A w(Y_1, …, Y_n) where A is change of basis matrix and det A is always positive (a continuous real-valued nowhere vanishing function on a connected set must have constant sign: its image must be a connected subset of R- {0}). it suffices to show that w(Y_1, … Y_n) has constant sign. Suppose U has coordinate r_1, …, r_n. Then w = f dr1 … drn. As above, f must have constant sign. Let B be the change of coordinate matrix between Y_i and d/dr_i. Then w(Y_1, …, Y_n) = f (det B) w(d/dr_1, …, d/dr_n) = f (det B) must have constant sign. Conversely suppose that dr_1…dr_n(X1, .., Xn) is positive on U. Then We claim that d/dr_1,…, d/dr_n is a continuous representative of X_1, .., X_n on U. This is because 0 < dr_1…dr_n(X1, .., Xn) = det C dr_1…dr_n(d/dr_1, .., d/dr_n) = det C, where C is the change of basis matrix.
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