29 a if we replace r by y this becomes x2 n2 y 0 145 y

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: that ∂2u ∂ ∂u ∂u ∂ ∂ ∂u ( )= cos θ + sin θ = cos θ = 2 ∂r ∂r ∂r ∂r ∂x ∂y ∂r = cos2 θ ∂u ∂x + sin θ ∂ ∂r ∂u ∂y ∂2u ∂2u ∂2u + 2 cos θ sin θ + sin2 θ 2 . ∂x2 ∂x∂y ∂y Similarly, ∂2u ∂ ∂u ∂u ∂ ∂u ( )= −r sin θ + r cos θ = ∂θ2 ∂θ ∂θ ∂θ ∂x ∂y = −r sin θ ∂ ∂θ = r2 sin2 θ ∂u ∂x + r cos θ ∂ ∂θ ∂u ∂y − r cos θ ∂u ∂u − r sin θ ∂x ∂y ∂2u ∂2u ∂2u ∂u − 2r2 cos θ sin θ + r2 cos2 θ 2 − r , 2 ∂x ∂x∂y ∂y ∂r which yields 1 ∂2u ∂2u ∂2u ∂ 2 u 1 ∂u + cos2 θ 2 − . = sin2 θ 2 − 2 cos θ sin θ r2 ∂θ2 ∂x ∂x∂y ∂y r ∂r Adding these results together, we obtain ∂2u 1 ∂2u ∂ 2 u ∂ 2 u 1 ∂u + 2 2= + 2− , 2 ∂r r ∂θ ∂x2 ∂y r ∂r or equivalently, ∆u = ∂2u 1 ∂ 2 u 1 ∂u . + 2 2+ 2 ∂r r ∂θ r ∂r 137 Finally, we can write this result in the form ∆u = 1∂ r ∂r r ∂u ∂r + 1 ∂2u . r2 ∂θ2 (5.24) This formula for the Laplace operator, together wit...
View Full Document

Ask a homework question - tutors are online