Math

Advanced MathQ&A Library11. (Euler's Theorem on Homogeneous Functions) We say f: R"- {0} R is homogeneous of degree k if f(tx) = tf(x) for all t >0. Prove that f is homogeneous of degree k if and only if Df (x)x= kf(x) for all nonzero x e R". (Hint: Fix x and consider h(t) = 1-*f (tx) .) 12. Suppose UCR is open and convex (i.e., given any points a, b e U, the line segment joining them lies in U as well). If f: U -R is differentiable and Df (x) = O for all x e U, prove that f is constant. Can you prove this when U is open and connected (i.e., any pair of points can be joined by piecewise-C path)? a 13. Suppose f: R > R is differentiable and let h fxy2) for x 0. Letting r = /x2 +y2, show that ahe he =rf(r). x dx 14. Suppose h: R - R is continuous and u, v: (a, b) tion F: (a, b) -R given by R are differentiable. Prove that the func- h(s)ds F(t)= is differentiable and calculate F'. (Hint: Recall that the Fundamental Theorem of Calculus tells you how to differentiate functions such as H (r) = 2Start your trial now! First week only $4.99!*arrow_forward*

Question

Hello,

Please help me solve number 13 in the attachment.

Thank you very much.

1 Rating

Tagged in

Math

Advanced Math

Find answers to questions asked by students like you.

Q: Abstract Algebra

A: To define the concept of a subfield of a field and prove the stated property regarding subfields of ...

Q: Suppose that f : R → R is differentiable and define g(x) = x^3 f(x^3). Show that g is differentiable...

A: Given:The function f : R → R is differentiable.The function g(x) = x3 f(x3).

Q: For problems 1 through 11, use the following sets. The set U is the universal set. 0- -32, - 8, -2, ...

A: Hey, since there are multiple questions posted, we will answer first question. If you want any speci...

Q: 3. In how many ways can you arrange 5 books on a shelf from available 10?

A: The number of all combination of n things, taken r at time is given by:

Q: Use the Pe sidue Theoem and the fact 4ha

A: Click to see the answer

Q: Question attached. Z is a complex variable, z=x+iy

A: To verify the Laplace equation (harmonicity condition ) for the given function and to find its harmo...

Q: Let i, j be coprime positive integers, and let R be an integral domain. Prove that (r-y) is a prime ...

A: To show that the ideal generated by x^i-y^j under the given conditions is a prime ideal in the ring ...

Q: Using the method of undetermined coefficients, find the general solution to the differential equatio...

A: The given differential equation is y′″ - 3y″ - y = 4et ...... (1)

Q: Use the method of integrating factors to solve the initial value problem. y'+xy=x, y(0)=1

A: Click to see the answer