Measure Theory : Continity and the Monotone Convergence Theorem

Let f be a nonnegative integrable function. Show that the function F defined by ... is continuous by using the Monotone Convergence Theorem. Please see the attached file for the fully formatted problems.

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

Important Formulas and their Explanations (I): Gradient, Divergence and Curl Gradient of the sum of two scalar point functions. ...continues

Gradient of the difference of two scalar point functions If f and g are two scalar point functions, then prove that grad ( f – g ) = grad f – grad g that is, ( f – g ) = f – g that is, gradient of ( f – g ) = gradient of f – gradient of g

Important Formulas and their Explanations (II): Gradient, Divergence and Curl Gradient of the differnece of two scalar point functions. ...continues

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 ...continues

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 ...continues

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 ...continues

Real Analysis : Limits and Cluster Points

Please see the attached file for the fully formatted problems. 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 o ...continues

Bounded Function : Signed Baire Measure

Show that each bounded function F of bounded variation gives rise to a finite signed Baire measure v such that v (a,b] = F(b+) minus F(a+)

Lebesgue-Stieltjes Integral

Please see attached

Weak Convergence

Please explain why the following sequence for otherwise is an example of a sequence in such that weakly, but not strongly. Please see the attached file for the fully formatted problem.