Consider the polar coordinate on R^2,i.e
x= rcosθ ; y = rsinθ
Compute the following partial derivatives in term of coordinate (r,θ)
a) ∂/∂x=?
b) ∂/∂y=?
c) ((∂^2)/(∂x^2))+((∂^2)/(∂y^2))=?
I know ∂/∂x= (∂/∂r)(∂r/∂x)+(∂/∂θ)(∂θ/∂x)
Is ∂r/∂x = cosθ and ∂θ/∂x = -rsinθ ?
Partial derivative...?
2015-05-17 2:26 am
回答 (1)
2015-05-18 8:07 pm
r = (x^2 + y^2)^1/2
dr/dx = x/(x^2+ y^2)^1/2 = x/r
dr/dy = y/r
theta = tan^-1 y/x
dt/dx = 1/(1+(y/x)^2) (-y/x^2)
= -y/(x^2 + y^2)
= -y/r^2
dt/dy = 1/(1+(y/x)^2) (1/x)
dt/dy = x/(x^2+y^2)
= x/r^2
dr/dx = x/(x^2+ y^2)^1/2 = x/r
dr/dy = y/r
theta = tan^-1 y/x
dt/dx = 1/(1+(y/x)^2) (-y/x^2)
= -y/(x^2 + y^2)
= -y/r^2
dt/dy = 1/(1+(y/x)^2) (1/x)
dt/dy = x/(x^2+y^2)
= x/r^2
收錄日期: 2021-04-15 20:11:47
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