2) Prove that: a) dx' dx- b) F) = dF %3D Ldx dx c) [x,p²]= 2h?d/dx

Show that t, e^t, and sin(t) are linearly independent.

5. Chose the curl of ƒ (x, y, z) = x²î + xyzĵ– zk at the point (2, 1, -2). a) 2î + 2k b)-2i-2j c) 4î-4j+2k d) - 2i-2k

Show that for any vector u in R^n, (u^(T))*(u) = 0 if and only if u = 0.

Please help with steps!

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3. Let V denote the vector space of all functions f : R" → R, equipped with addition + : V × V → V defined via (ƒ+g)(x) = f(x)+ g(x), x ≤ R", and scalar multiplication : R × V → V defined via (\ · ƒ)(x) = \ƒ(x), ● x ER". Rº Now let W = {ƒ : R^ → R : ƒ(x) = ax + b for some a, b € R}, i.e. the space of all linear functions R" → R. (a) Find a basis for W. You should prove that it is indeed a basis.

Let w = x" dz A dx + (y° + x°) dx ^ dy be a two-form on R³. Determine the one-form *d(*w), where * is the Hodge star operator in IR°. dz dx + |dy+ *d(*w)