Chapter 6: Applications of Double Integration
Section 6.3: Surface Area
Derive the expression for dσ when the surface is given parametrically. See Table 6.3.2.
The position-vector form for the parametrically given surface is
R=xu,v i+yu,v j+zu,v k
Represented as position vectors, the following are coordinate curves on the surface along which v=b and u=a, respectively.
Ru=xu,b i+yu,b j+zu,b k and Rv=xa,v i+ya,v j+za,v k
Vectors tangent to these curves are
Tu=xu i+yu j+zu k and Tv=xv i+yv j+zv k
A vector normal to the surface at a,b is then N=Tu×Tv, that is,
= yuzuyvzv| i−|xuzuxvzv j+|xuyuxvyv| k
= yuyvzuzv| i−|xuxvzuzv j+|xuxvyuyv| k
= yuyvzuzv| i+|zuzvxuxv j+|xuxvyuyv| k
= ∂y,z∂u,v i+∂z,x∂u,v j+∂x,y∂u,v k
= J1 i+J2 j+J3 k
where the determinant representing the cross product has been first-row expanded. In the second row of the display, the property that a determinant does not change if the array is transposed is used. In the third row of the display, the property that a determinant changes sign if two rows are interchanged is used.
The length of N is then N=J12+J22+J32 = λ, so dσ=λ dA^.
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