Chapter 6: Applications of Double Integration
Section 6.3: Surface Area
Derive the expression for dσ when the surface is given explicitly. See Table 6.3.2.
The surface-area element dσ is actually the area of an "infinitesimal" parallelogram attached to the surface and formed by to "infinitesimal" tangent vectors along two intersecting coordinate curves. If the surface is represented via the position vector R, then Rx dx and Ry dy will be infinitesimal tangent vectors along the coordinate curves Rx,a and Rb,y, respectively. The cross product of these tangent vectors is normal to the surface, and its length is the area of the infinitesimal parallelogram formed by these tangent vectors at a,b. Table 6.3.14(a) summarizes these relationships.
Rx dx=10fx dx
Ry dy=01fy dy
N=Rx dx×Ry dy=−fx−fy1 dA
Table 6.3.14(a) An infinitesimal normal N on a surface defined by the position vector R
The surface-area element dσ is the length of N, so dσ=N = 1+fx2+fy2 dA.
Maple Solution - Interactive
Student Multivariate Calculus
Context Panel: Assign Name
Obtain vectors tangent to coordinate curves
Calculus palette: Partial derivative operator
Context Panel: Evaluate and Display Inline
Context Panel: Assign to a Name≻Rx (then Ry)
∂∂ x R = 10∂∂x⁢f⁡x,y→assign to a nameRx
∂∂ y R = 01∂∂y⁢f⁡x,y→assign to a nameRy
Common Symbols palette:
Type the norm bars or get them from the Operators palette or the Layout palette.
N = 1+∂∂x⁢f⁡x,y2+∂∂y⁢f⁡x,y2
Maple Solution - Coded
Since dσ=λ dA, the calculations in Table 6.3.14(b) lead to λ=1+fx2+fy2. In the Cartesian coordinates assumed in Table 6.3.2, dA will have the form dy dx or dx dy. The surface-area element is actually the area of an "infinitesimal" parallelogram attached to the surface and formed by to "infinitesimal" tangent vectors along two intersecting coordinate curves. The difference between an infinitesimal tangent vector and a finite one is just the differential factor (either dx or dy). Hence, the calculations in Table 6.3.14(b) omit these differentials, and lead to λ rather than dσ.
Install the Student MultivariateCalculus package.
Define the surface via the position vector R.
Obtain N, normal to the surface, and its length
Use the diff command to obtain vectors tangent to each coordinate curve.
Use the CrossProduct command to obtain N as the cross product of two tangent vectors.
Use the Norm command to obtain the (Euclidean) length of N.
NormN = 1+∂∂x⁢f⁡x,y2+∂∂y⁢f⁡x,y2
Table 6.3.14(b) Calculating λ=1+fx2+fy2 for a surface given explicitly by z=fx,y
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