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Consider the function z = f(x, y) implicitly defined by Þ(x, y, z) = x³ + y³ - 2² + cos(xz) = 0. a) Use the explicit function theorem to decide whether z = f(x,y) exists and is differentiable in some neighbourhood of (ro, yo) = (0-1) such that 0= f(0, -1) and (ro, yo) = (0, 1) such that √2 = f(0, 1). b) Restrictz to z> 0 and calculate the partial derivatives af ər (0,2) af ди and c) Assume r and y to be functions of the new variables u, v in the form x(u, v) = u²-v² 2 Calculate the partial derivatives |(u=1,v=1) af ду (0,2) and and y(u, v) = 3uv. af อง |(u=1,v=1)

Consider the function z = f(x, y) implicitly defined by Þ(x, y, z) = x³ + y³ - 2² + cos(xz) = 0. a) Use the explicit function theorem to decide whether z = f(x,y) exists and is differentiable in some neighbourhood of (ro, yo) = (0-1) such that 0= f(0, -1) and (ro, yo) = (0, 1) such that √2 = f(0, 1). b) Restrictz to z> 0 and calculate the partial derivatives af ər (0,2) af ди and c) Assume r and y to be functions of the new variables u, v in the form x(u, v) = u²-v² 2 Calculate the partial derivatives |(u=1,v=1) af ду (0,2) and and y(u, v) = 3uv. af อง |(u=1,v=1)

Advanced Engineering Mathematics
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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Consider the function z = f(x, y) implicitly defined by (x, y, z) = x³ +y³ - 2²+ cos(xz) = 0. a) Use the explicit function theorem to decide whether z = f(x,y) exists and is differentiable in some neighbourhood of (ro, yo) (0-1) such that 0 = f(0, -1) and (ro, yo) (0, 1) such that √2 = f(0,1). b) Restrictz to z> 0 and calculate the partial derivatives af ər (0,2) af ди and c) Assume r and y to be functions of the new variables u, v in the form x(u, v) = u²-v² 2 Calculate the partial derivatives (u=1,v=1) af ду (0,2) and and y(u, v) = 3uv. af Əv |(u=1,v=1)
Transcribed Image Text:Consider the function z = f(x, y) implicitly defined by (x, y, z) = x³ +y³ - 2²+ cos(xz) = 0. a) Use the explicit function theorem to decide whether z = f(x,y) exists and is differentiable in some neighbourhood of (ro, yo) (0-1) such that 0 = f(0, -1) and (ro, yo) (0, 1) such that √2 = f(0,1). b) Restrictz to z> 0 and calculate the partial derivatives af ər (0,2) af ди and c) Assume r and y to be functions of the new variables u, v in the form x(u, v) = u²-v² 2 Calculate the partial derivatives (u=1,v=1) af ду (0,2) and and y(u, v) = 3uv. af Əv |(u=1,v=1)
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