Explanation Given information: Δ φ = ∂ 2 φ ∂ x 2 + ∂ 2 φ ∂ y 2 + ∂ 2 φ ∂ z 2 , where Δ denotes the Laplace operator. Concept used: Let the vector field F → = A i → + B j → + C k → then the divergence is defined to be, Div F → = ∂ A ∂ x + ∂ B ∂ y + ∂ C ∂ z Proof: Let φ ( x , y , z ) be the vector field So, the divergence is to be: Div φ = ∇ φ = ∂ ∂ x i ^ + ∂ ∂ y j ^ + ∂ ∂ z z ^ If again find the divergence of ∇ φ then: ∇ φ = ∂ φ ∂ x + ∂ φ ∂ y + ∂ φ ∂ z Div ( ∇ φ ) = ( ∂ φ ∂ x + ∂ φ ∂ y + ∂ φ ∂ z ) ( ∂ φ ∂ x + ∂ φ ∂ y + ∂ φ ∂ z ) Div ( ∇ φ ) = ( ∂ φ ∂ x ∂ φ ∂ x + ∂ φ ∂ x

4th Edition

ISBN: 9781319545956

Author: Rogawski

Publisher: MAC HIGHER

See similar textbooks

Videos

Question

Chapter 17.3, Problem 35E

To determine

Prove that div(∇φ)=Δφ if Δφ=∂2φ∂x2+∂2φ∂y2+∂2φ∂z2, where Δ denotes the Laplace operator.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 17.3, Problem 34E

Chapter 17.3, Problem 36E

Students have asked these similar questions

How to find the radius of convergence for the series in the image below? I'm stuck on how to isolate the x in the interval of convergence.

Determine the exact signed area between the curve g(x): x-axis on the interval [0,1]. = tan2/5 secx dx and

Set up the partial fraction expansion of the function below. Do not explicitly solve for the variables 5 x²(x − 2)(x − 3)³ (24 - 81)² -

Chapter 17 Solutions

CALCULUS W/ACHIEVE PLATFORM

Ch. 17.1 - Prob. 1PQ Ch. 17.1 - Prob. 2PQ Ch. 17.1 - Prob. 3PQ Ch. 17.1 - Prob. 4PQ Ch. 17.1 - Prob. 5PQ Ch. 17.1 - Prob. 1E Ch. 17.1 - Prob. 2E Ch. 17.1 - Prob. 3E Ch. 17.1 - Prob. 4E Ch. 17.1 - Prob. 5E

Ch. 17.1 - Prob. 6E Ch. 17.1 - Prob. 7E Ch. 17.1 - Prob. 8E Ch. 17.1 - Prob. 9E Ch. 17.1 - Prob. 10E Ch. 17.1 - Prob. 11E Ch. 17.1 - Prob. 12E Ch. 17.1 - Prob. 13E Ch. 17.1 - Prob. 14E Ch. 17.1 - Prob. 15E Ch. 17.1 - Prob. 16E Ch. 17.1 - Prob. 17E Ch. 17.1 - Prob. 18E Ch. 17.1 - Prob. 19E Ch. 17.1 - Prob. 20E Ch. 17.1 - Prob. 21E Ch. 17.1 - Prob. 22E Ch. 17.1 - Prob. 23E Ch. 17.1 - Prob. 24E Ch. 17.1 - Prob. 25E Ch. 17.1 - Prob. 26E Ch. 17.1 - Prob. 27E Ch. 17.1 - Prob. 28E Ch. 17.1 - Prob. 29E Ch. 17.1 - Prob. 30E Ch. 17.1 - Prob. 31E Ch. 17.1 - Prob. 32E Ch. 17.1 - Prob. 33E Ch. 17.1 - Prob. 34E Ch. 17.1 - Prob. 35E Ch. 17.1 - Prob. 36E Ch. 17.1 - Prob. 37E Ch. 17.1 - Prob. 38E Ch. 17.1 - Prob. 39E Ch. 17.1 - Prob. 40E Ch. 17.1 - Prob. 41E Ch. 17.1 - Prob. 42E Ch. 17.1 - Prob. 43E Ch. 17.1 - Prob. 44E Ch. 17.1 - Prob. 45E Ch. 17.1 - Prob. 46E Ch. 17.1 - Prob. 47E Ch. 17.1 - Prob. 48E Ch. 17.1 - Prob. 49E Ch. 17.1 - Prob. 50E Ch. 17.1 - Prob. 51E Ch. 17.2 - Prob. 1PQ Ch. 17.2 - Prob. 2PQ Ch. 17.2 - Prob. 3PQ Ch. 17.2 - Prob. 4PQ Ch. 17.2 - Prob. 5PQ Ch. 17.2 - Prob. 1E Ch. 17.2 - Prob. 2E Ch. 17.2 - Prob. 3E Ch. 17.2 - Prob. 4E Ch. 17.2 - Prob. 5E Ch. 17.2 - Prob. 6E Ch. 17.2 - Prob. 7E Ch. 17.2 - Prob. 8E Ch. 17.2 - Prob. 9E Ch. 17.2 - Prob. 10E Ch. 17.2 - Prob. 11E Ch. 17.2 - Prob. 12E Ch. 17.2 - Prob. 13E Ch. 17.2 - Prob. 14E Ch. 17.2 - Prob. 15E Ch. 17.2 - Prob. 16E Ch. 17.2 - Prob. 17E Ch. 17.2 - Prob. 18E Ch. 17.2 - Prob. 19E Ch. 17.2 - Prob. 20E Ch. 17.2 - Prob. 21E Ch. 17.2 - Prob. 22E Ch. 17.2 - Prob. 23E Ch. 17.2 - Prob. 24E Ch. 17.2 - Prob. 25E Ch. 17.2 - Prob. 26E Ch. 17.2 - Prob. 27E Ch. 17.2 - Prob. 28E Ch. 17.2 - Prob. 29E Ch. 17.2 - Prob. 30E Ch. 17.2 - Prob. 31E Ch. 17.2 - Prob. 32E Ch. 17.2 - Prob. 33E Ch. 17.2 - Prob. 34E Ch. 17.2 - Prob. 35E Ch. 17.2 - Prob. 36E Ch. 17.2 - Prob. 37E Ch. 17.2 - Prob. 38E Ch. 17.3 - Prob. 1PQ Ch. 17.3 - Prob. 2PQ Ch. 17.3 - Prob. 3PQ Ch. 17.3 - Prob. 4PQ Ch. 17.3 - Prob. 5PQ Ch. 17.3 - Prob. 1E Ch. 17.3 - Prob. 2E Ch. 17.3 - Prob. 3E Ch. 17.3 - Prob. 4E Ch. 17.3 - Prob. 5E Ch. 17.3 - Prob. 6E Ch. 17.3 - Prob. 7E Ch. 17.3 - Prob. 8E Ch. 17.3 - Prob. 9E Ch. 17.3 - Prob. 10E Ch. 17.3 - Prob. 11E Ch. 17.3 - Prob. 12E Ch. 17.3 - Prob. 13E Ch. 17.3 - Prob. 14E Ch. 17.3 - Prob. 15E Ch. 17.3 - Prob. 16E Ch. 17.3 - Prob. 17E Ch. 17.3 - Prob. 18E Ch. 17.3 - Prob. 19E Ch. 17.3 - Prob. 20E Ch. 17.3 - Prob. 21E Ch. 17.3 - Prob. 22E Ch. 17.3 - Prob. 23E Ch. 17.3 - Prob. 24E Ch. 17.3 - Prob. 25E Ch. 17.3 - Prob. 26E Ch. 17.3 - Prob. 27E Ch. 17.3 - Prob. 28E Ch. 17.3 - Prob. 29E Ch. 17.3 - Prob. 30E Ch. 17.3 - Prob. 31E Ch. 17.3 - Prob. 32E Ch. 17.3 - Prob. 33E Ch. 17.3 - Prob. 34E Ch. 17.3 - Prob. 35E Ch. 17.3 - Prob. 36E Ch. 17.3 - Prob. 37E Ch. 17.3 - Prob. 38E Ch. 17.3 - Prob. 39E Ch. 17.3 - Prob. 40E Ch. 17.3 - Prob. 41E Ch. 17.3 - Prob. 42E Ch. 17.3 - Prob. 43E Ch. 17.3 - Prob. 44E Ch. 17 - Prob. 1CRE Ch. 17 - Prob. 2CRE Ch. 17 - Prob. 3CRE Ch. 17 - Prob. 4CRE Ch. 17 - Prob. 5CRE Ch. 17 - Prob. 6CRE Ch. 17 - Prob. 7CRE Ch. 17 - Prob. 8CRE Ch. 17 - Prob. 9CRE Ch. 17 - Prob. 10CRE Ch. 17 - Prob. 11CRE Ch. 17 - Prob. 12CRE Ch. 17 - Prob. 13CRE Ch. 17 - Prob. 14CRE Ch. 17 - Prob. 15CRE Ch. 17 - Prob. 16CRE Ch. 17 - Prob. 17CRE Ch. 17 - Prob. 18CRE Ch. 17 - Prob. 19CRE Ch. 17 - Prob. 20CRE Ch. 17 - Prob. 21CRE Ch. 17 - Prob. 22CRE Ch. 17 - Prob. 23CRE Ch. 17 - Prob. 24CRE Ch. 17 - Prob. 25CRE Ch. 17 - Prob. 26CRE Ch. 17 - Prob. 27CRE Ch. 17 - Prob. 28CRE Ch. 17 - Prob. 29CRE Ch. 17 - Prob. 30CRE Ch. 17 - Prob. 31CRE Ch. 17 - Prob. 32CRE Ch. 17 - Prob. 33CRE Ch. 17 - Prob. 34CRE Ch. 17 - Prob. 35CRE Ch. 17 - Prob. 36CRE Ch. 17 - Prob. 37CRE Ch. 17 - Prob. 38CRE Ch. 17 - Prob. 39CRE Ch. 17 - Prob. 40CRE Ch. 17 - Prob. 41CRE

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.

Prove that div( ∇ φ ) = Δ φ if Δ φ = ∂ 2 φ ∂ x 2 + ∂ 2 φ ∂ y 2 + ∂ 2 φ ∂ z 2 , where Δ denotes the Laplace operator.

Solution Summary: The author explains how the divergence of phi (x,y,z) is defined.

Videos

Prove that div(∇φ)=Δφ if Δφ=∂2φ∂x2+∂2φ∂y2+∂2φ∂z2, where Δ denotes the Laplace operator.

Want to see the full answer?

Chapter 17 Solutions