Jacobian Determinants & Coordinate Transformations: Solved Exam Questions + Formula Cheat Sheet

JACOBIAN: KEY FORMULAS CHEAT SHEET

1. Standard Definition (2 Variables) If u = f (x, y) and
v = g(x, y), the Jacobian is:
∂u ∂u
∂(u, v) ∂x ∂y
=
∂(x, y) ∂v ∂v
∂x ∂y

2. General Definition (n Variables) For functions u1, u2, . . . , un
of variables x1 , x2 , . . . , xn :
∂u1 ∂u1
∂x1 ... ∂xn
∂(u1 , u2 , . . . , un ) .. ... ..
= . .
∂(x1 , x2 , . . . , xn )
∂un ∂un
∂x1 ... ∂xn


3. Chain Rule (Function of Functions) If u, v are functions
of r, s and r, s are functions of x, y:
∂(u, v) ∂(u, v) ∂(r, s)
= ×
∂(x, y) ∂(r, s) ∂(x, y)

4. Inverse Property

∂(u, v) ∂(x, y)
× =1
∂(x, y) ∂(u, v)

5. Implicit Functions For equations F1 = 0, F2 = 0, . . . , Fn = 0:

∂(F1 ,...,Fn )
∂(u1 , . . . , un ) ∂(x ,...,x )
= (−1)n ∂(F1 ,...,Fn )
∂(x1 , . . . , xn ) 1 n
∂(u1 ,...,un )




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, 6. Important Standard Results
A. Polar Coordinates (2D) Relation: x = r cos θ, y = r sin θ

∂(x, y)
J= =r
∂(r, θ)

B. Cylindrical Coordinates (3D) Relation: x = r cos θ, y =
r sin θ, z = z
∂(x, y, z)
J= =r
∂(r, θ, z)
C. Spherical Coordinates (3D) Relation:

x = r sin θ cos ϕ
y = r sin θ sin ϕ
z = r cos θ

∂(x, y, z)
J= = r2 sin θ
∂(r, θ, ϕ)




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