Gradient of a constant If c(x,y,z) is a constant, then Grad c = 0, which is a zero vector, that is, gradient of the constant c = 0, which is a zero vector, that is, c = 0 , which is a zero vector.

Important Formulas and their Explanations (III): Gradient, Divergence and Curl Gradient of a constant Gradient of a constant If c(x,y,z) is a constant, then grad c = 0, which is a zero vector, that is, gradient of the constant c = 0, which is a zero vector, that is, c = 0 , which is a zero vector. ... click for more

Gradient of the product of two scalar point functions. If f and g are two scalar point functions, then grad (fg) = f grad g + g grad f, that is, gradient of (fg) = f gradient of g + g gradient of f, that is, ( fg ) = f g + g f

Important Formulas and their Explanations (IV): Gradient, Divergence and Curl Gradient of the product of two scalar point functions ... click for more

Gradient of the quotient of two scalar point functions. If f and g are two scalar point functions, then grad (f/g) = ( g grad f – f grad g )/g^2, that is, gradient of (f/g) = ( g gradient of f – f gradient of g )/g^2, that is, (f/g) = (g f – f g)/g^2

Important Formulas and their Explanations (V): Gradient, Divergence and Curl Gradient of the quotient of two scalar point functions ... click for more

Real Analysis

(See attached file for full problem description) --- 1) Show that if xn > 0 for all n in the natural numbers., then lim(xn) = 0 if and only if lim(1/xn) = +∞. (Note: xn is a sequence) 2) Let Σan be a given series and let Σbn be the series in which the terms are the same and in the same order as in Σan except that the terms for which an = 0 have been omitted. Show that Σan converges to A if and only if Σbn converges to A. 3) Let c be a cluster point of and let . Prove that if and only if 4) Let I be an interval in , let f : I → , and let . Suppose there exist constants K and L such that for . ... click for more

Real Analysis

(See attached file for full problem description) --- 1) Prove that does not exist but that . 2) Let f, g be defined on to , and let c be a cluster point of A. Suppose that f is bounded on a neighborhood of c and that . Prove that . 3) Let f, g be defined on A to and let c be a cluster point of A. (a) Show that if both and exist, then exists. (b) If and exist, does it follow that exists? 4) Let f : be such that f(x+y)=f(x)+f(y) for all x, y in . Assume that exists. Prove that L = 0, and then prove that f has a limit at every point . (Hint: First note that f(2x) = f(x)+f(x) = 2f(x) for . Also note that f(x) = f(x - c) + f(c) for x,... click for more