Explanation The given function is, v ( r , s , t ) = r cos ( s + 2 t ) . Differentiate the given function with respect to r and obtain v r . ∂ v ∂ r = ∂ ∂ r ( r cos ( s + 2 t ) ) = cos ( s + 2 t ) ∂ ∂ r ( r ) = cos ( s + 2 t ) ( 1 ) ∂ v ∂ r = cos ( s + 2 t ) (1) Differentiate the equation (1) with respect to r and obtain the second order derivative, v r r . ∂ 2 v ∂ r 2 = ∂ ∂ r ( cos ( s + 2 t ) ) = 0 Thus, v r r ( r , s , t ) = 0 . Differentiate the given function with respect to sand obtain v s . ∂ v ∂ s = ∂ ∂ s ( r cos ( s + 2 t ) ) = r ∂ ∂ s ( cos ( s + 2 t ) ) = r [ − sin ( s + 2 t ) ( 1 + 0 ) ] ∂ v ∂ s = − r sin ( s + 2 t ) . (2) Differentiate the equation (2) with respect to s and obtain the second order derivative, v s s . ∂ 2 v ∂ s 2 = ∂ ∂ s ( − r sin ( s + 2 t ) ) = − r ∂ ∂ s ( sin ( s + 2 t ) ) = − r [ cos ( s + 2 t ) ( 1 + 0 ) ] = − r cos ( s + 2 t ) Hence, v s s ( r , s , t ) = − r cos ( s + 2 t ) . Differentiate the given function with respect to t and obtain v t . ∂ v ∂ t = ∂ ∂ t ( r cos ( s + 2 t ) ) = r ∂ ∂ t ( cos ( s + 2 t ) ) = r [ − sin ( s + 2 t ) ( 0 + 2 ) ] ∂ v ∂ t = − 2 r sin ( s + 2 t ) (3) Differentiate the equation (3) with respect to t and obtain the second order derivative, v t t . ∂ 2 v ∂ t 2 = ∂ ∂ t ( − 2 r sin ( s + 2 t ) ) = − 2 r ∂ ∂ t ( sin ( s + 2 t ) ) = − 2 r [ cos ( s + 2 t ) ( 0 + 2 ) ] = − 4 r cos ( s + 2 t ) Hence, v t t ( r , s , t ) = − 4 r cos ( s + 2 t ) . Differentiate equation (1) with respect to s and obtain the partial derivative, v r s . ∂ 2 v ∂ r ∂ s = ∂ ∂ s ( cos ( s + 2 t ) ) = − sin ( s + 2 t ) ( 1 + 0 ) = − sin ( s + 2 t ) Therefore, v r s ( r , s , t ) = − sin ( s + 2 t ) . Differentiate equation (2) with respect to r and obtain the partial derivative, v s r . ∂ 2 v ∂ s ∂ r = ∂ ∂ r ( − r sin ( s + 2 t ) ) = − sin ( s + 2 t ) ∂ ∂ r ( r ) = − sin ( s + 2 t ) ( 1 ) = − sin ( s + 2 t ) Thus, v s r ( r , s , t ) = − sin ( s + 2 t )

Essential Calculus: Early Transcendentals; MAC 2311 Sequence| MAC 2281Sequence USF (Essential Calculus)

2nd Edition

ISBN: 9781285101552

Author: James Stewart

Publisher: Cenage

See similar textbooks

Question

Chapter 11, Problem 20RE

To determine

To find: The second order partial derivative of the function v(r,s,t)=rcos(s+2t) .

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 11, Problem 19RE

Chapter 11, Problem 21RE

Students have asked these similar questions

Use Laplace transform to solve the initial value problem y' + y = tsin(t), y(0) = 0

The function g is defined by g(x) = sec² x + tan x. What are all solutions to g(x) = 1 on the interval 0 ≤ x ≤ 2π ? A x = = 0, x == = 3, x = π, x = 7 4 , 4 and x 2π only = B x = 4' 1, x = 1, x = 57 and x = 3 only C x = πk and x = - +πk D , where is any integer П x = +πk and П x = +πk, where k is any integer

Vector v = PQ has initial point P (2, 14) and terminal point Q (7, 3). Vector v = RS has initial point R (29, 8) and terminal point S (12, 17). Part A: Write u and v in linear form. Show all necessary work. Part B: Write u and v in trigonometric form. Show all necessary work. Part C: Find 7u − 4v. Show all necessary calculations.

Chapter 11 Solutions

Essential Calculus: Early Transcendentals; MAC 2311 Sequence| MAC 2281Sequence USF (Essential Calculus)

Ch. 11.1 - Let g(x, y) = cos(x + 2y). (a) Evaluate g(2, 1)....Ch. 11.1 - Let F(x,y)=1+4y2. (a) Evaluate F(3,1). (b) Find...Ch. 11.1 - Let f(x,y,z)=x+y+z+ln(4x2y2z2). (a) Evaluate f(1,...Ch. 11.1 - Let g(x,y,z)=x3y2z10xyz. (a) Evaluate g(1, 2, 3)....Ch. 11.1 - Find and sketch the domain of the function....Ch. 11.1 - Find and sketch the domain of the function....Ch. 11.1 - Find and sketch the domain of the function. 15....Ch. 11.1 - Find and sketch the domain of the function....Ch. 11.1 - Find and sketch the domain of the function. 19....Ch. 11.1 - Find and sketch the domain of the function. f(x,...

Knowledge Booster

The second order partial derivative of the function v ( r , s , t ) = r cos ( s + 2 t ) .

To find: The second order partial derivative of the function v(r,s,t)=rcos(s+2t) .

Want to see the full answer?

Chapter 11 Solutions