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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
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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### 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 \\ &=
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 \\ &=
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