# Question: Let f a b x c d R be continuous

Let f: [a, b] x [c, d] → R be continuous and suppose D 2 f is continuous. Define f (y) = ∫ ba f (x, y) dx. Prove Leibnitz' Rule: f1 (y) = ∫ ba D2 f (x, y) dx.

## Answer to relevant Questions

If f: [a, b] x [c, d] → R is continuous and D2f is continuous, define F (x, y) = ∫xa (t,y) dt a. Find D1F and D2F. (b) If G (x) = ∫ g(x) f (t, x) dt, find G1 (x).Use Theorem 3-14 to prove Theorem 3-13 without the assumption that g1 (x) ≠ 0.If f: Rn → Rn, the graph of f is {(x, y): y = f (x)}. Show that the graph of is an -dimensional manifold if and only if is differentiable.If M C R n is an orientable (n - 1)-dimensional manifold, show that there is an open set A C Rn and a differentiable g: A→ R1 so that M = g-1 (0) and g1 (x) has rank 1 for x ЄM.If M is an oriented one-dimensional manifold in RN and c: [0, 1] →M is orientation-preserving, show thatPost your question