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Prove the following. Let M C R3 be a smooth surface, let Z, Y:M R3 be smooth vector fields, let f:M R be a smooth
Prove the following. Let M C R3 be a smooth surface, let Z, Y:M R3 be smooth vector fields, let f:M R be a smooth function, let p E M be a point, let u, w E TM be vectors and let a, b be real numbers. Then (i) au-buz = M,+ b,Z. (ii) u(Z+Y) = ,Z+VY (iii) uf Z = (uf)Z(p) + f(p)(,Z). (iv) VO{Z, Y) = (V,Z,Y(p)) + (Z(p), ,Y). Prove the following. Let M C R3 be a smooth surface, let Z, Y:M R3 be smooth vector fields, let f:M R be a smooth function, let p E M be a point, let u, w E TM be vectors and let a, b be real numbers. Then (i) au-buz = M,+ b,Z. (ii) u(Z+Y) = ,Z+VY (iii) uf Z = (uf)Z(p) + f(p)(,Z). (iv) VO{Z, Y) = (V,Z,Y(p)) + (Z(p), ,Y)
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