Parameterization helps us describe 3D surfaces using 2D parameters. We can map points on a plane to points on a surface, making it easier to analyze complex shapes. This technique is crucial for understanding the geometry of curved surfaces in space.
Surface area calculations build on parameterization. By using vector-valued functions and partial derivatives , we can find the area of curved surfaces through integration. This connects our understanding of surfaces to calculus concepts we've learned before.
Parameterization and Surface Area
Parameterization of 3D surfaces
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Vector-valued functions for surfaces map 2D parameters to 3D points r ( u , v ) = ⟨ x ( u , v ) , y ( u , v ) , z ( u , v ) ⟩ \mathbf{r}(u,v) = \langle x(u,v), y(u,v), z(u,v) \rangle r ( u , v ) = ⟨ x ( u , v ) , y ( u , v ) , z ( u , v )⟩ defined over a domain in uv-plane
Common surface types with parameterizations:
Planes: r ( u , v ) = p + u v + v w \mathbf{r}(u,v) = \mathbf{p} + u\mathbf{v} + v\mathbf{w} r ( u , v ) = p + u v + v w where p \mathbf{p} p is a point on the plane and v \mathbf{v} v , w \mathbf{w} w are vectors in the plane
Spheres : r ( θ , ϕ ) = ⟨ a sin ϕ cos θ , a sin ϕ sin θ , a cos ϕ ⟩ \mathbf{r}(\theta,\phi) = \langle a\sin\phi\cos\theta, a\sin\phi\sin\theta, a\cos\phi \rangle r ( θ , ϕ ) = ⟨ a sin ϕ cos θ , a sin ϕ sin θ , a cos ϕ ⟩ with a a a as radius, θ \theta θ as azimuthal angle, ϕ \phi ϕ as polar angle
Cylinders : r ( θ , z ) = ⟨ a cos θ , a sin θ , z ⟩ \mathbf{r}(\theta,z) = \langle a\cos\theta, a\sin\theta, z \rangle r ( θ , z ) = ⟨ a cos θ , a sin θ , z ⟩ where a a a is radius and z z z is height
Cones : r ( θ , z ) = ⟨ z tan α cos θ , z tan α sin θ , z ⟩ \mathbf{r}(\theta,z) = \langle z\tan\alpha\cos\theta, z\tan\alpha\sin\theta, z \rangle r ( θ , z ) = ⟨ z tan α cos θ , z tan α sin θ , z ⟩ with α \alpha α as half-angle of cone
Identifying parameters involves recognizing surface symmetry and natural coordinates (radial, angular)
Transforming between Cartesian and parametric forms requires understanding geometric relationships and solving equations
Surface area of parametric surfaces
Surface integral calculates area by summing infinitesimal patches over entire surface
Fundamental formula S A = ∬ D ∣ r u × r v ∣ d u d v SA = \iint_D |\mathbf{r}_u \times \mathbf{r}_v| \, du \, dv S A = ∬ D ∣ r u × r v ∣ d u d v uses partial derivatives and cross product
Surface area calculation process:
Determine parameterization r ( u , v ) \mathbf{r}(u,v) r ( u , v ) based on surface geometry
Compute partial derivatives r u \mathbf{r}_u r u and r v \mathbf{r}_v r v using component-wise differentiation
Calculate cross product r u × r v \mathbf{r}_u \times \mathbf{r}_v r u × r v representing surface normal vector
Find magnitude ∣ r u × r v ∣ |\mathbf{r}_u \times \mathbf{r}_v| ∣ r u × r v ∣ using dot product or component formula
Set up double integral over parameter domain D and evaluate using appropriate techniques (substitution, numerical methods)
Jacobian matrix J = [ ∂ x ∂ u ∂ x ∂ v ∂ y ∂ u ∂ y ∂ v ∂ z ∂ u ∂ z ∂ v ] J = \begin{bmatrix} \frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} \\ \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} \\ \frac{\partial z}{\partial u} & \frac{\partial z}{\partial v} \end{bmatrix} J = ∂ u ∂ x ∂ u ∂ y ∂ u ∂ z ∂ v ∂ x ∂ v ∂ y ∂ v ∂ z encodes information about surface deformation
Relationship r u × r v = det ( J ) \mathbf{r}_u \times \mathbf{r}_v = \det(J) r u × r v = det ( J ) connects cross product to Jacobian determinant
Derivation steps:
Approximate surface with small parallelograms formed by r u Δ u \mathbf{r}_u \Delta u r u Δ u and r v Δ v \mathbf{r}_v \Delta v r v Δ v
Area of parallelogram given by magnitude of cross product: ∣ r u × r v ∣ Δ u Δ v |\mathbf{r}_u \times \mathbf{r}_v| \Delta u \Delta v ∣ r u × r v ∣Δ u Δ v
Take limit as Δ u , Δ v → 0 \Delta u, \Delta v \to 0 Δ u , Δ v → 0 to get infinitesimal area element
Express result using Jacobian: d A = det ( J T J ) d u d v dA = \sqrt{\det(J^T J)} \, du \, dv d A = det ( J T J ) d u d v
Final surface area formula S A = ∬ D det ( J T J ) d u d v SA = \iint_D \sqrt{\det(J^T J)} \, du \, dv S A = ∬ D det ( J T J ) d u d v generalizes cross product form
Surface area in coordinate systems
Cartesian coordinates for surfaces z = f ( x , y ) z = f(x,y) z = f ( x , y ) :
Parameterization r ( x , y ) = ⟨ x , y , f ( x , y ) ⟩ \mathbf{r}(x,y) = \langle x, y, f(x,y) \rangle r ( x , y ) = ⟨ x , y , f ( x , y )⟩ uses x, y as parameters
Surface area formula S A = ∬ D 1 + ( ∂ f ∂ x ) 2 + ( ∂ f ∂ y ) 2 d x d y SA = \iint_D \sqrt{1 + (\frac{\partial f}{\partial x})^2 + (\frac{\partial f}{\partial y})^2} \, dx \, dy S A = ∬ D 1 + ( ∂ x ∂ f ) 2 + ( ∂ y ∂ f ) 2 d x d y derived from general form
Spherical coordinates :
Conversion x = ρ sin ϕ cos θ , y = ρ sin ϕ sin θ , z = ρ cos ϕ x = \rho\sin\phi\cos\theta, y = \rho\sin\phi\sin\theta, z = \rho\cos\phi x = ρ sin ϕ cos θ , y = ρ sin ϕ sin θ , z = ρ cos ϕ with ρ \rho ρ as radius, θ \theta θ as azimuthal angle, ϕ \phi ϕ as polar angle
Surface area element d S = ρ 2 sin ϕ d θ d ϕ dS = \rho^2\sin\phi \, d\theta \, d\phi d S = ρ 2 sin ϕ d θ d ϕ useful for surfaces with spherical symmetry
Cylindrical coordinates :
Conversion x = r cos θ , y = r sin θ , z = z x = r\cos\theta, y = r\sin\theta, z = z x = r cos θ , y = r sin θ , z = z with r r r as radial distance, θ \theta θ as angular coordinate
Surface area element d S = r d θ d z dS = r \, d\theta \, dz d S = r d θ d z simplifies calculations for cylindrical surfaces
Choosing coordinate system based on surface symmetry reduces complexity of integration
Changing variables in surface integrals involves Jacobian of transformation between coordinate systems