Determine whether each of the following is true or false. (a) Any two distinct non-parallel lines...

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Determine whether each of the following is true or false. (a) Any two distinct non-parallel lines in R³ must have an inter- section point. (b) Let f,g: R → R be two smooth functions. Define the smooth function h: R2 → R by h(x, y) = f(x)g(y). If P₁(x) and Q₁ (y) are the first-order Taylor's polynomials of f(x) and g(y) using 0 as the reference point respectively, then the second-order Taylor's polynomial of h(x, y) using (0,0) as the reference point must be P₁(x)Q₁(y). (c) Define F: R³ → R³ by F(x, y, z) = (z + sin x, y cos x, yz). Then F is not the curl of any vector field in R³ of class C². (d) Let C be the line segment in R2 joining the points (0,0) and (1,1). Then there are exactly two different parametrizations of C. Determine whether each of the following is true or false. (a) Any two distinct non-parallel lines in R³ must have an inter- section point. (b) Let f,g: R → R be two smooth functions. Define the smooth function h: R2 → R by h(x, y) = f(x)g(y). If P₁(x) and Q₁ (y) are the first-order Taylor's polynomials of f(x) and g(y) using 0 as the reference point respectively, then the second-order Taylor's polynomial of h(x, y) using (0,0) as the reference point must be P₁(x)Q₁(y). (c) Define F: R³ → R³ by F(x, y, z) = (z + sin x, y cos x, yz). Then F is not the curl of any vector field in R³ of class C². (d) Let C be the line segment in R2 joining the points (0,0) and (1,1). Then there are exactly two different parametrizations of C.

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