1. f(x,y)=x²+y2 is a homogenous fur ion of degree 22. Euler's theorem is: x +y =Iəzдzəx ду3. A function is said to be homogenous of degree n if it can bewritten in the form x"0), where can be any function.4. Mixed derivatives theorem says that fxy (a, b) = fyx (a,b)5.Second order partial derivative of Z with respect to x if z= exy isx² exyActivate Winden

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Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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D. Let z = f(x, y) = VI+2y + I a) Compute the local linearization (z = L(r, y)) of f at (0,0). b) Quadratic approximation (z = Q(r, y)) to.f centered at (0,0). c) Compute the linear and quadratic approximation to f(0.1,0.15) (to 4 decimal places) and %3D compare to the actual value.
7. Let F(x, y) = 2x²yi + 3xyj, and C be the boundary of the square with corner points (±1, +1), positively oriented. Find the flux of F across C. (12 points) -1)→ y=1 from (-1) 101=-xdx 3
a) Find the global maximum and minimum for the function x – xy + y in the square (both inside and the boundary) with vertices (-2, –2), (-2,0), (0,0), (0, –2). b) Compute the differential dz and first order partials z,, ze for a composite function z r cos(0), y = r sin(0). xy +x2, x =
6) Let F(x, y) = [sin(y) - y² + 1] x cos(y) - 2xy Is F conservative? If so, give a potential function for F.
5. Let consider f(x, y) = x³ + 4xy – 2y² + 1. (a) Find the gradient at A(1,1). (b) Find the Hessian of f at A. (c) Find the second-order Taylor approximation for f at A.
Let F = (x'e* +2xy – 2 y)i+(cos y +x² +3x) j and let C be the ellipse with equation (2x – 2 y)² + (x+ y)² = 9 . Find the work (circulation) associated with F along the curve C traversed counterclockwise. [Hint: You may have to invoke two important theorems to complete this problem.]
5. Evaluate the path integral [1 F. dr where F(x, y, z) = (3x²yz − 3y)i + (x³z − 3x)j + (x³y + 2z)k and C is the helix with initial point (0, 1,) and terminal point (-1,0, π) defined by r(t) = cos(t), sin(t), t>. (538831889)
#22. Let fexiys= Show that of ах x²y4 x+6y8 if (x,y) (0,0) if (x,y) = (0,0) of dy (0,0) and (0,0) exist b.) Show that f is not differentiable at co,os by showing that f is not continous at (0,0).

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1. f(x,y)=x²+y² is a homogenous furion of degree 2