What is a triple integral?
I = ∭V f(x, y, z) dV where V is the region/domain of integration and dV is the infinitesimal volume element.
What is the volume element in Cartesian coordinates?
dV = dx dy dz
How is a point represented in cylindrical coordinates?
P = (R, φ, z), where (R, φ) is the projection onto the xy-plane and z is the height.
Conversion formulas between Cartesian and cylindrical coordinates
x = R cos φ, y = R sin φ, z = z R² = x² + y², tan φ = y/x
Examples of surfaces in cylindrical coordinates
Cylinder: x² + y² = a² → R = a
Cone: az = √(x² + y²) → R = az
Paraboloid: x² + y² = a²z → R = a√z
Hyperboloid: x² + y² − z² = a² → R² = a² + z²
Triple integrals in cylindrical coordinates
∭V f(x, y, z) dV = ∭ f(R cos φ, R sin φ, z) dV with dV = R dR dφ dz
How is a point represented in spherical coordinates?
P = (r, θ, φ)
r ≥ 0 = distance from origin
θ ∈ [0, π] = angle from positive z-axis
φ ∈ [0, 2π) = angle in xy-plane
Conversion formulas between Cartesian and spherical coordinates
x = r sin θ cos φ
y = r sin θ sin φ
z = r cos θ
Triple integrals in spherical coordinate
dV = r² sin θ dr dθ dφ
What is the Jacobian determinant for (u, v, w) → (x, y, z)?
J(u, v, w) =
| ∂x/∂u ∂x/∂v ∂x/∂w |
| ∂y/∂u ∂y/∂v ∂y/∂w |
| ∂z/∂u ∂z/∂v ∂z/∂w |
Change of variables formula for triple integrals
∭V f(x, y, z) dx dy dz = ∭S f(x(u, v, w), y(u, v, w), z(u, v, w)) |J(u, v, w)| du dv dw
