Prove the following. Let M C R be a smooth surface, let Z, Y: M ...

   

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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 function, let p E M be a point, let u, w E TM be vectors and let a, b be real numbers. Then (i) ▼av+bwZ = aỸ„Z + bỸ„Z. (ii) ▼₂(Z+Y) = V₂Z + V₂Y (iii) ▼vƒZ = (Ñ‚ƒ)Z(p) + ƒ(p)(ỸvZ). (iv) Ỹ₂(Z,Y) = (Ỹ„Z,Y(p)) + (Z(p), Ñ„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) ▼av+bwZ = aỸ„Z + bỸ„Z. (ii) ▼₂(Z+Y) = V₂Z + V₂Y (iii) ▼vƒZ = (Ñ‚ƒ)Z(p) + ƒ(p)(ỸvZ). (iv) Ỹ₂(Z,Y) = (Ỹ„Z,Y(p)) + (Z(p), Ñ„Y).

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avbw Z avZbvwZ Since Z is a smooth vector field we have Zp ZpZp where Zp is the tangent vector and Z... View the full answer