Question

A function f”R n × R m → R is bilinear if for all x, y...

A function f”R n × R m → R is bilinear if for all x, y ∈ R n and all w, z ∈ R m, and all a ∈ R: • f(x + ay, z) = f(x, z) + af(y, z) • f(x, w + az) = f(x, w) + af(x, z) (a) Prove that if f is bilinear, then (0.1) lim (h,k)→(0,0) |f(h, k)| |(h, k)| = 0. (b) Prove that Df(a, b) · (h, k) = f(a, k) + f(h, y).

Math   Advanced-Math   
0 0
Add a commentTranscribed image text
Next >

Homework Answers

Answer #1

0 0
Know the answer?
Your Answer:

Post as a guest

Your Name:

What's your source?

Earn Coins

Coins can be redeemed for fabulous gifts.

Log In

Sign Up

Not the answer you're looking for?
Ask your own homework help question
Similar Questions
Is the function f(x,y)=x−yx+y continuous at the point (−1,−1)? If not, why is the function not...
Is the function f(x,y)=x−yx+y continuous at the point (−1,−1)? If not, why is the function not continuous? Select the correct answer below: A. Yes B. No, because lim(x,y)→(−1,1)x−yx+y=−1 and f(0,0)=0. C. No, because lim(x,y)→(−1,1)x−yx+y does not exist and f(0,0) does not exist. D. No, because lim(x,y)→(0,0)x2−y2x2+y2=1 and f(0,0)=0.
Let f : R → R be defined by f(x) = x^3 + 3x, for all...
Let f : R → R be defined by f(x) = x^3 + 3x, for all x. (i) Prove that if y > 0, then there is a solution x to the equation f(x) = y, for some x > 0. Conclude that f(R) = R. (ii) Prove that the function f : R → R is strictly monotone. (iii) By (i)–(ii), denote the inverse function (f ^−1)' : R → R. Explain why the derivative of the inverse function,...
when z=f(x,y), where tan(xyz)=x+y+z, find az/ax and az/ay
when z=f(x,y), where tan(xyz)=x+y+z, find az/ax and az/ay
Suppose that f : R → R such that, the lim h→0 [f(x) − f(x −...
Suppose that f : R → R such that, the lim h→0 [f(x) − f(x − h)] = 0 for all x ∈ R, then is f continuous in this case?
1. For n exists in R, we define the function f by f(x)=x^n, x exists in...
1. For n exists in R, we define the function f by f(x)=x^n, x exists in (0,1), and f(x):=0 otherwise. For what value of n is f integrable? 2. For m exists in R, we define the function g by g(x)=x^m, x exists in (1,infinite), and g(x):=0 otherwise. For what value of m is g integrable?
If f:X→Y is a function and A⊆X, then define f(A) ={y∈Y:f(a) =y for some a∈A}. (a)...
If f:X→Y is a function and A⊆X, then define f(A) ={y∈Y:f(a) =y for some a∈A}. (a) If f:R→R is defined by f(x) =x^2, then find f({1,3,5}). (b) If g:R→R is defined by g(x) = 2x+ 1, then find g(N). (c) Suppose f:X→Y is a function. Prove that for all B, C⊆X,f(B∩C)⊆f(B)∩f(C). Then DISPROVE that for all B, C⊆X,f(B∩C) =f(B)∩f(C).
1. A function f : Z → Z is defined by f(n) = 3n − 9....
1. A function f : Z → Z is defined by f(n) = 3n − 9. (a) Determine f(C), where C is the set of odd integers. (b) Determine f^−1 (D), where D = {6k : k ∈ Z}. 2. Two functions f : Z → Z and g : Z → Z are defined by f(n) = 2n^ 2+1 and g(n) = 1 − 2n. Find a formula for the function f ◦ g. 3. A function f :...
For each vector field F~ (x, y) = hP(x, y), Q(x, y)i, find a function f(x,...
For each vector field F~ (x, y) = hP(x, y), Q(x, y)i, find a function f(x, y) such that F~ (x, y) = ∇f(x, y) = h ∂f ∂x , ∂f ∂y i by integrating P and Q with respect to the appropriate variables and combining answers. Then use that potential function to directly calculate the given line integral (via the Fundamental Theorem of Line Integrals): a) F~ 1(x, y) = hx 2 , y2 i Z C F~ 1...
hi guys , using this definition for limits in higher dimensions : lim (x,y)→(a,b) f(x, y)...
hi guys , using this definition for limits in higher dimensions : lim (x,y)→(a,b) f(x, y) = L if 1. ∃r > 0 s.th. f(x, y) is defined when 0 < || (x, y) − (a, b) || < r and 2. given ε > 0 we can find δ > 0 s.th. 0 < || (x, y) − (a, b) || < δ =⇒ | f(x, y) − L | < ε how do i show that this is...
2. Let A = {p, q, r, s}, B = {k, l, m, n}, and C...
2. Let A = {p, q, r, s}, B = {k, l, m, n}, and C = {u, v, w}, Define f : A→B by f(p) = m, f(q) = k, f(r) = l, and f(s) = n, and define g : B→C by g(k) = v, g(l) = w, g(m) = u, and g(n) = w. Also define h : A→C by h = g ◦ f. (a) Write out the values of h. (b) Why is it that...
ADVERTISEMENT
Need Online Homework Help?

Get Answers For Free
Most questions answered within 1 hours.

Ask a Question
ADVERTISEMENT