(a) To determine The value of ∂ u ∂ x , ∂ u ∂ y and ∂ u ∂ z as function of x , y and z . Express u directly in terms of x , y and z before differentiating. Explanation Given information: u = p − q q − r , p = x + y + z , q = x − y + z , r = x + y − z ; ( x , y , z ) = ( 3 , 2 , 1 ) . Formula: d d x ( u v ) = v u ' − u v ' v 2 Calculation: Express the function as below. u = p − q q − r (1) Partial Differentiate with respect to p in equation (1). ∂ u ∂ p = 1 q − r Partial Differentiate with respect to q in equation (1). ∂ u ∂ q = ( q − r ) ( − 1 ) − ( p − q ) ( 1 ) ( q − r ) 2 = − q + r − p + q ( q − r ) 2 = r − p ( q − r ) 2 Partial Differentiate with respect to r in the equation (1). ∂ u ∂ r = ( p − q ) ( − 1 ( q − r ) 2 ( − 1 ) ) = p − q ( q − r ) 2 Express the value of p as below. p = x + y + z Partial Differentiate with respect to x . ∂ p ∂ x = 1 Express the value of q as below. q = x − y + z Partial Differentiate with respect to x . ∂ q ∂ x = 1 Express the value of r as below. r = x + y − z Partial Differentiate with respect to x . ∂ r ∂ x = 1 Determine the value of ∂ u ∂ x . Apply chain rule. ∂ u ∂ x = ∂ u ∂ p ∂ p ∂ x + ∂ u ∂ q ∂ q ∂ x + ∂ u ∂ r ∂ r ∂ x Substitute 1 q − r for ∂ u ∂ p , r − p ( q − r ) 2 for ∂ u ∂ q , p − q ( q − r ) 2 for ∂ u ∂ r , 1 for ∂ p ∂ x , 1 for ∂ q ∂ x and 1 for ∂ r ∂ x . ∂ u ∂ x = 1 q − r ( 1 ) + r − p ( q − r ) 2 ( 1 ) + p − q ( q − r ) 2 ( 1 ) = 1 q − r + r − p ( q − r ) 2 + p − q ( q − r ) 2 = q − r + r − p + p − q ( q − r ) 2 = 0 Express the value of p as below. p = x + y + z Partial Differentiate with respect to y . ∂ p ∂ y = 1 Express the value of q as below. q = x − y + z Partial Differentiate with respect to y . ∂ q ∂ y = − 1 Express the value of r as below. r = x + y − z Partial Differentiate with respect to y . ∂ r ∂ y = 1 Determine the value of ∂ u ∂ y . Apply chain rule. ∂ u ∂ y = ∂ u ∂ p ∂ p ∂ y + ∂ u ∂ q ∂ q ∂ y + ∂ u ∂ r ∂ r ∂ y Substitute 1 q − r for ∂ u ∂ p , r − p ( q − r ) 2 for ∂ u ∂ q , p − q ( q − r ) 2 for ∂ u ∂ r , 1 for ∂ p ∂ y , -1 for ∂ q ∂ y and 1 for ∂ r ∂ y . ∂ u ∂ y = 1 q − r ( 1 ) + r − p ( q − r ) 2 ( − 1 ) + p − q ( q − r ) 2 ( 1 ) = 1 q − r − r − p ( q − r ) 2 + p − q ( q − r ) 2 = q − r − r + p + p − q ( q − r ) 2 = 2 p − 2 r ( q − r ) 2 Substitute x + y + z for p , x − y + z for q and x + y − z for r (b) To determine The value of ∂ u ∂ x , ∂ u ∂ y and ∂ u ∂ z at the given point of ( x , y , z ) .

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ISBN: 9780134654874

Author: WEIR

Publisher: VST

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Chapter 14.4, Problem 11E

(a)

To determine

The value of ∂u∂x, ∂u∂y and ∂u∂z as function of x, y and z.

Express u directly in terms of x, y and z before differentiating.

(b)

To determine

The value of ∂u∂x, ∂u∂y and ∂u∂z at the given point of (x,y,z).

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Chapter 14.4, Problem 10E

Chapter 14.4, Problem 12E

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The value of ∂ u ∂ x , ∂ u ∂ y and ∂ u ∂ z as function of x , y and z . Express u directly in terms of x , y and z before differentiating.