Question

express ∂w\∂r and ∂w\∂s in terms of r and s if w=x^2 + y^2, x=r - s, y=r + s.

Answer 1

Step-by-step explanation:

w = x² + y²

x = r - s

y = r + s

w = (r - s) ² + (r + s) ² = r² + s² - 2rs + r² +s² + 2rs = 2r² + 2s

w = 2r² + 2s²

∂w / ∂r = ∂(2r² + 2s²) / ∂r = [ 2 ∂(r²) / ∂r] + [ 2 ( ∂(s²) / ∂s ) ( ∂s / ∂r ) ]

= 4r + 4s (∂s / ∂r)

∂w / ∂s = ∂(2r² + 2s²) / ∂s = [ 2 ( ∂(r²) / ∂r ) ( ∂r / ∂s ) ] + [ 2 ∂(s²) / ∂s ]

= 4r (∂r / ∂s) + 4s