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^.
<< Previous Example Section 6.3
Next Section >>
© Maplesoft, a division of Waterloo Maple Inc., 2021. All rights reserved. This product is protected by copyright and distributed under licenses restricting its use, copying, distribution, and decompilation.
For more information on Maplesoft products and services, visit www.maplesoft.com
Download Help Document
What kind of issue would you like to report? (Optional)