Differentiation

Domain, Range & Function Graphs Rectangular Coordinates Cylindrical Coordinates Parametric Equations Contents

[D]

Slice Curves in Spherical Coordinates Rectangular Coordinates Cylindrical Coordinates Parametric Equations Top of Page Contents

For every point (r₀,θ₀,φ₀) in the domain of a function f, the intersection of the graph of f with the vertical plane r = r₀, θ = θ₀ will be the (r₀,θ₀)-slice curve (r₀,θ₀,φ,f(r₀,θ₀,φ)). The domain of the r₀-slice curve is the set of φ for which (r₀,θ₀,φ) is in the domain of f.

Similarly we define the (θ₀,φ₀)-slice curve to be (r,θ₀,φ₀,f(r,θ₀,φ₀)) for all r such that (r,θ₀,φ₀) is in the domain of f, and we define the (r₀,φ₀)-slice curve to be (r₀,θ,φ₀,f(r₀,θ,φ₀)) for all θ such that (r₀,θ,φ₀) is in the domain of f.

[D]

The slice curves with θ = θ₀ and φ = φ₀ give all lines through the point with spherical coordinates {r₀,θ₀,φ₀}.

Slice Surfaces in Spherical Coordinates Rectangular Coordinates Cylindrical Coordinates Parametric Equations Top of Page Contents

For every point (r₀,θ₀,φ₀) in the domain of a function f, the intersection of the graph of f with the vertical hyperplane φ = φ₀, will be the φ₀-slice surface (r,θ,φ₀,f(r,θ,φ₀)). The domain of the φ₀-slice surface is the set of (r, θ) for which (r,θ,φ₀) is in the domain of f.

Similarly we define the θ₀-slice surface to be (r,θ₀,φ,f(r,θ₀,φ)) for all (r, φ) such that (r,θ₀,φ) is in the domain of f, and we define the r₀-slice surface to be (r₀,θ,φ,f(r₀,θ,φ)) for all (θ, φ) such that (r₀,θ,φ) is in the domain of f.

[D]

Level Sets and Contours in Spherical Coordinates Rectangular Coordinates Cylindrical Coordinates Top of Page Contents

[D]