Homework Help is Here – Start Your Trial Now!
Math
Advanced Math
1. Consider the function f defined by f(x, y) = x²y + y²x x² + y² 0 if (x, y) (0,0); " if (x, y)=(0,0). (a) Show that f is continuous at (0,0). (Hint: Evaluate the appro- priate limit by changing to polar coordinates.) (b) Using the principal definition of partial derivatives, show that (0,0) exists and equals 0. af მე (c) You may assume that af მყ (0, 0) exists and is equal to 0. Using the definition of differentiablity of a multivariable function, explain why f is not differentiable at (0,0). (Hint: Investigate the exis- tence or otherwise of the appropriate limit by changing to polar coordinates.) 2. Consider the region R in the xy-plane given by R={(x,y) R² x ≥ 0, y ≥ 0 and 1 ≤ x² + y² ≤ 2}. (a) Sketch the region R. (b) By changing to polar coordinates evaluate the double integral y sin (π (x² + y²)) √x² + y² dx dy. Recall that Cartesian and polar coordinates are related by (x, y) = (r cos 0, r sin 0), ✓ (1) where r = √√√x² + y², and 0 ≤ 0 < 27. You may assume that the Jacobian for the change of variables (1) is given by (x, y) ǝ (r, 0) = 1.

1. Consider the function f defined by f(x, y) = x²y + y²x x² + y² 0 if (x, y) (0,0); " if (x, y)=(0,0). (a) Show that f is continuous at (0,0). (Hint: Evaluate the appro- priate limit by changing to polar coordinates.) (b) Using the principal definition of partial derivatives, show that (0,0) exists and equals 0. af მე (c) You may assume that af მყ (0, 0) exists and is equal to 0. Using the definition of differentiablity of a multivariable function, explain why f is not differentiable at (0,0). (Hint: Investigate the exis- tence or otherwise of the appropriate limit by changing to polar coordinates.) 2. Consider the region R in the xy-plane given by R={(x,y) R² x ≥ 0, y ≥ 0 and 1 ≤ x² + y² ≤ 2}. (a) Sketch the region R. (b) By changing to polar coordinates evaluate the double integral y sin (π (x² + y²)) √x² + y² dx dy. Recall that Cartesian and polar coordinates are related by (x, y) = (r cos 0, r sin 0), ✓ (1) where r = √√√x² + y², and 0 ≤ 0 < 27. You may assume that the Jacobian for the change of variables (1) is given by (x, y) ǝ (r, 0) = 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
See similar textbooks
icon
Related questions
Question
1. Consider the function f defined by f(x, y) = x²y + y²x x² + y² 0 if (x, y) (0,0); " if (x, y)=(0,0). (a) Show that f is continuous at (0,0). (Hint: Evaluate the appro- priate limit by changing to polar coordinates.) (b) Using the principal definition of partial derivatives, show that (0,0) exists and equals 0. af მე (c) You may assume that af მყ (0, 0) exists and is equal to 0. Using the definition of differentiablity of a multivariable function, explain why f is not differentiable at (0,0). (Hint: Investigate the exis- tence or otherwise of the appropriate limit by changing to polar coordinates.) 2. Consider the region R in the xy-plane given by R={(x,y) R² x ≥ 0, y ≥ 0 and 1 ≤ x² + y² ≤ 2}. (a) Sketch the region R. (b) By changing to polar coordinates evaluate the double integral y sin (π (x² + y²)) √x² + y² dx dy. Recall that Cartesian and polar coordinates are related by (x, y) = (r cos 0, r sin 0), ✓ (1) where r = √√√x² + y², and 0 ≤ 0 < 27. You may assume that the Jacobian for the change of variables (1) is given by (x, y) ǝ (r, 0) = 1.
Transcribed Image Text:1. Consider the function f defined by f(x, y) = x²y + y²x x² + y² 0 if (x, y) (0,0); " if (x, y)=(0,0). (a) Show that f is continuous at (0,0). (Hint: Evaluate the appro- priate limit by changing to polar coordinates.) (b) Using the principal definition of partial derivatives, show that (0,0) exists and equals 0. af მე (c) You may assume that af მყ (0, 0) exists and is equal to 0. Using the definition of differentiablity of a multivariable function, explain why f is not differentiable at (0,0). (Hint: Investigate the exis- tence or otherwise of the appropriate limit by changing to polar coordinates.) 2. Consider the region R in the xy-plane given by R={(x,y) R² x ≥ 0, y ≥ 0 and 1 ≤ x² + y² ≤ 2}. (a) Sketch the region R. (b) By changing to polar coordinates evaluate the double integral y sin (π (x² + y²)) √x² + y² dx dy. Recall that Cartesian and polar coordinates are related by (x, y) = (r cos 0, r sin 0), ✓ (1) where r = √√√x² + y², and 0 ≤ 0 < 27. You may assume that the Jacobian for the change of variables (1) is given by (x, y) ǝ (r, 0) = 1.
Expert Solution
steps

Step by step

Solved in 2 steps

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Similar questions
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
Basic Technical Mathematics
Basic Technical Mathematics
Advanced Math
ISBN:
9780134437705
Author:
Washington
Publisher:
PEARSON
Topology
Topology
Advanced Math
ISBN:
9780134689517
Author:
Munkres, James R.
Publisher:
Pearson,
SEE MORE TEXTBOOKS