MAC2313 Study Guide - Quiz Guide: Divergence Theorem, Cross Product, Simply Connected Space

1 Divergence The divergence of a vector field F as in (1) is defined as dív F =-+-+- All of the terms in (2) are functions of (r,y, z), which we have omitted to keep the notation brief. Also note that while F is a vector function, its divergence div F is a simple function of three variables, sometimes called a scalar field (as opposed to a vector field). In situations where F(x, y,z) describes the flow of a physical quantity div F can be interpreted as the local flur of the field, measuring how much of the quantity is flowing away from the point (r, y,z) (which is the case if div F(x, y, z) > 0) or into the point (if div F(x, y, z) 0)

Let r R and For which values r R is the set {u. v. w} linearly independent? For which values r R is the vector b a linear combination of u, v and w? For which of these values of r can b be written as a linear combination of u. v and w in more than one way? The set R of all column vectors of length three, with real entries, is a vector space. Is the subset a subspace of R3? Justify your answer. Show that the set of all twice differentiable functions f : R rightarrow R satisfying the differential equation sin (x)f"(x) + x2f(x) = 0 is a vector space with respect to the usual operations of addition of functions and multiplication by scalars. Here, f" denotes the second derivative of f. Let S be the following subset of the vector space P3 of all real polynomials p of degree at most 3: S = {p P3 | p(1) = 0, p'(1) = 1}, where p' is the derivative of p. Determine whether S is a subspace of P3. Determine whether the polynomial q(x) = x - 2x2 + x3 is an element of S.

Let r R and For which values r R is the set {u, v, w} linearly independent? For which values r R is the vector b a linear combination of u, v and w? For which of these values of r can b be written as a linear combination of u, v and w in more than one way? The set R3 of all column vectors of length three, with real entries, is a vector space. Is the subset a subspace of R3? Justify your answer. Show that the set of all twice differentiable functions f : R rightarrow R satisfying the differential equation sin (x)f"(x) + x2f(x) = 0 is a vector space with respect to the usual operations of addition of functions and multiplication by scalars. Here f" denotes the second derivative of f. Let S be the following subset of the vector space P3 of all real polynomials p of degree at most where p' is the derivative of p. Determine whether S is a subspace of P3. Determine whether the polynomial q(x) = x - 2x2 + x3 is an element of S.