31. a. A torus of revolution (doughnut) is obtained by rotating a circle C in the xz-plane about the z-axis in space. (See the accompa- nying figure.) If C has radius r > 0 and center (R, 0, 0), show...

31. a. A torus of revolution (doughnut) is obtained by rotating a circle<br>C in the xz-plane about the z-axis in space. (See the accompa-<br>nying figure.) If C has radius r > 0 and center (R, 0, 0), show<br>that a parametrization of the torus is<br>r(и, v)<br>((R + r cos u)cos v)i<br>+ ((R + r cos u)sin v)j + (r sin u)k,<br>where 0 < u < 27 and 0 < v < 2m are the angles in the<br>figure.<br>b. Show that the surface area of the torus is A = 47²Rr.<br>х<br>R -<br>r(и, v)<br>

Extracted text: 31. a. A torus of revolution (doughnut) is obtained by rotating a circle C in the xz-plane about the z-axis in space. (See the accompa- nying figure.) If C has radius r > 0 and center (R, 0, 0), show that a parametrization of the torus is r(и, v) ((R + r cos u)cos v)i + ((R + r cos u)sin v)j + (r sin u)k, where 0 < u="">< 27="" and="" 0="">< v="">< 2m="" are="" the="" angles="" in="" the="" figure.="" b.="" show="" that="" the="" surface="" area="" of="" the="" torus="" is="" a="47²Rr." х="" r="" -="" r(и,="">

Jun 05, 2022

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31. a. A torus of revolution (doughnut) is obtained by rotating a circle C in the xz-plane about the z-axis in space. (See the accompa- nying figure.) If C has radius r > 0 and center (R, 0, 0), show...

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