W(s, t) = F(u(s, t), v(s, t)), where F, u, and v are differentiable. If u(3, 6) = -7, us (3,- 6) = 9, ut (3, 6) = 3, v(3, — 6) = 7, vs(3, — 6) = — 2, vt (3,6)= 5, Fu(-7, 7) -4, and F(-7, 7) = -3, then find the following: W,(3, 6) Wt(3, 6) =
W(s, t) = F(u(s, t), v(s, t)), where F, u, and v are differentiable. If u(3, 6) = -7, us (3,- 6) = 9, ut (3, 6) = 3, v(3, — 6) = 7, vs(3, — 6) = — 2, vt (3,6)= 5, Fu(-7, 7) -4, and F(-7, 7) = -3, then find the following: W,(3, 6) Wt(3, 6) =
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![### Problem Statement
Consider the function \( W(s,t) = F(u(s,t), v(s,t)) \), where \( F \), \( u \), and \( v \) are differentiable. Given the following information:
- \( u(3, -6) = -7 \)
- \( u_s(3, -6) = 9 \)
- \( u_t(3, -6) = 3 \)
- \( v(3, -6) = 7 \)
- \( v_s(3, -6) = -2 \)
- \( v_t(3, -6) = 5 \)
- \( F_u(-7, 7) = -4 \)
- \( F_v(-7, 7) = -3 \)
Find the partial derivatives \( W_s(3, -6) \) and \( W_t(3, -6) \).
### Partial Derivatives Calculation
To find the partial derivatives \( W_s(s, t) \) and \( W_t(s, t) \) at the point \( (3, -6) \), we use the chain rule.
#### \( W_s(s, t) \) Calculation
\[ W_s(s, t) = F_u(u(s, t), v(s, t)) \cdot u_s(s, t) + F_v(u(s, t), v(s, t)) \cdot v_s(s, t) \]
Substituting the given values at \( (3, -6) \):
- \( u(3, -6) = -7 \)
- \( u_s(3, -6) = 9 \)
- \( v(3, -6) = 7 \)
- \( v_s(3, -6) = -2 \)
- \( F_u(-7, 7) = -4 \)
- \( F_v(-7, 7) = -3 \)
\[
\begin{align*}
W_s(3, -6) &= F_u(-7, 7) \cdot u_s(3, -6) + F_v(-7, 7) \cdot v_s(3, -6) \\
&= (-4) \cdot 9 + (-3) \cdot (-2) \\
&= -36 + 6 \\
&=](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F74b6836e-7075-4604-8b7e-b1441272473c%2F2bcaa337-d281-43b6-b5af-b27806ef1ba0%2Fr3m6dn_processed.png&w=3840&q=75)
Transcribed Image Text:### Problem Statement
Consider the function \( W(s,t) = F(u(s,t), v(s,t)) \), where \( F \), \( u \), and \( v \) are differentiable. Given the following information:
- \( u(3, -6) = -7 \)
- \( u_s(3, -6) = 9 \)
- \( u_t(3, -6) = 3 \)
- \( v(3, -6) = 7 \)
- \( v_s(3, -6) = -2 \)
- \( v_t(3, -6) = 5 \)
- \( F_u(-7, 7) = -4 \)
- \( F_v(-7, 7) = -3 \)
Find the partial derivatives \( W_s(3, -6) \) and \( W_t(3, -6) \).
### Partial Derivatives Calculation
To find the partial derivatives \( W_s(s, t) \) and \( W_t(s, t) \) at the point \( (3, -6) \), we use the chain rule.
#### \( W_s(s, t) \) Calculation
\[ W_s(s, t) = F_u(u(s, t), v(s, t)) \cdot u_s(s, t) + F_v(u(s, t), v(s, t)) \cdot v_s(s, t) \]
Substituting the given values at \( (3, -6) \):
- \( u(3, -6) = -7 \)
- \( u_s(3, -6) = 9 \)
- \( v(3, -6) = 7 \)
- \( v_s(3, -6) = -2 \)
- \( F_u(-7, 7) = -4 \)
- \( F_v(-7, 7) = -3 \)
\[
\begin{align*}
W_s(3, -6) &= F_u(-7, 7) \cdot u_s(3, -6) + F_v(-7, 7) \cdot v_s(3, -6) \\
&= (-4) \cdot 9 + (-3) \cdot (-2) \\
&= -36 + 6 \\
&=
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 2 steps with 1 images
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
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…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
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…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
