Explanation Given: Let’s consider, z = f ( r , θ ) Here, r and θ are defined as function of x and y by the equation x = r cos θ and y = r s i n θ Proof: Laplace equation in terms of the Cartesian coordinates system is, ∂ 2 z ∂ 2 x + ∂ 2 z ∂ 2 y = 0 The Cartesian coordinates can be represented by the polar coordinates as follows, x = r cos θ y = r s i n θ Let’s first compute the partial derivative of x , y with respect to r and θ ∂ x ∂ r = cos θ ∂ x ∂ θ = − r sin θ ∂ y ∂ r = sin θ ∂ y ∂ θ = r cos θ So first compute, ∂ z ∂ r for this, using the chain rule since ( x , y ) are function of ( r , θ ) ∂ z ∂ r = ∂ z ∂ x × ∂ x ∂ r + ∂ z ∂ y × ∂ y ∂ r = ∂ z ∂ x cos θ + ∂ z ∂ y sin θ = cos θ ∂ z ∂ x + sin θ ∂ z ∂ y Now calculate, ∂ 2 z ∂ 2 r ∂ 2 z ∂ 2 r = cos θ ∂ ∂ r ( ∂ z ∂ x ) + sin θ ∂ ∂ r ( ∂ z ∂ y ) = cos θ ( ∂ ∂ x ∂ z ∂ x ∂ x ∂ r + ∂ ∂ y ∂ z ∂ x ∂ y ∂ r ) + sin θ ( ∂ ∂ x ∂ z ∂ y ∂ x ∂ r + ∂ ∂ y ∂ z ∂ y ∂ y ∂ r ) = cos 2 θ ∂ 2 z ∂ 2 x 2 + 2 cos θ sin θ + ∂ 2 z ∂ x ∂ y + sin 2 θ ∂ 2 z ∂ 2 y 2 Similarly calculate, ∂ z ∂ θ and ∂ 2 z ∂ 2 θ ∂ z ∂ θ = ∂ z ∂ x × ∂ x ∂ θ + ∂ z ∂ y × ∂ y ∂ θ = ∂ z ∂ x ( − r sin θ ) + ∂ z ∂ y ( r cos θ ) = − r sin θ ∂ z ∂ x + r cos θ ∂ z ∂ y Now calculate, ∂ 2 z ∂ 2 θ ∂ 2 z ∂ 2 θ = − r cos θ ( ∂ z ∂ x ) + − r sin θ ∂ ∂ θ ( ∂ z ∂ x ) − r s i n θ ( ∂ z ∂ x ) + r s i n ∂ ∂ θ ( ∂ z ∂ y ) = − r cos θ ∂ z ∂ x − r sin θ ( ∂ ∂ x &

Calculus: Special Edition: Chapters 1-5 (w/ WebAssign)

6th Edition

ISBN: 9781524908102

Author: SMITH KARL J, STRAUSS MONTY J, TODA MAGDALENA DANIELE

Publisher: Kendall Hunt Publishing

See similar textbooks

Videos

Question

Chapter 11, Problem 86SP

To determine

To show: the equation ∂2z∂2x+∂2z∂2y=0 becomes ∂2z∂2r+1r2∂2z∂2θ+1r∂z∂r in polar coordinates

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 11, Problem 85SP

Chapter 11, Problem 87SP

Students have asked these similar questions

Please help me answer this. Show detailed and complete solution

The question is attached to this post. Please give full solution and explanation to the answer.

REFER TO IMAGE

Chapter 11 Solutions

Calculus: Special Edition: Chapters 1-5 (w/ WebAssign)

Ch. 11.1 - Prob. 1PS Ch. 11.1 - Prob. 2PS Ch. 11.1 - Prob. 3PS Ch. 11.1 - Prob. 4PS Ch. 11.1 - Prob. 5PS Ch. 11.1 - Prob. 6PS Ch. 11.1 - Prob. 7PS Ch. 11.1 - Prob. 8PS Ch. 11.1 - Prob. 9PS Ch. 11.1 - Prob. 10PS

Knowledge Booster

$Calculus$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.

the equation ∂ 2 z ∂ 2 x + ∂ 2 z ∂ 2 y = 0 becomes ∂ 2 z ∂ 2 r + 1 r 2 ∂ 2 z ∂ 2 θ + 1 r ∂ z ∂ r in polar coordinates

Solution Summary: The author illustrates the Laplace equation in terms of the Cartesian coordinates system by the equation x=rmathrmcostheta

Videos

To show: the equation ∂2z∂2x+∂2z∂2y=0 becomes ∂2z∂2r+1r2∂2z∂2θ+1r∂z∂r in polar coordinates

Want to see the full answer?

Chapter 11 Solutions