Calculus Volume 3 · 免费预览
柱坐标与球坐标下的三重积分
柱坐标在极坐标的基础上加入 z，体积元为 r dz dr d theta；球坐标使用 rho、theta、phi，体积元为 rho 的平方乘以 sin phi。两者都能大幅简化对称立体上的三重积分。
1. Cylindrical coordinates
Cylindrical coordinates
2. Spherical coordinates
Spherical coordinates
3. Choosing the system
Choosing the system
4. The Jacobian in triple form
The Jacobian in triple form
5. Cylindrical volume element
The cylindrical triple integral's volume element dV equals which expression?
6. Spherical ball limits
The ball of radius a in spherical coordinates has which limits?
7. Ball volume integral
The triple integral of 1 over the ball of radius a in spherical coordinates gives which volume?
8. When cylindrical wins
A solid consisting of a cylinder with a paraboloid's cap is most simply integrated in which coordinates?
9. Wrong system consequence
Using the wrong coordinate system for a triple integral typically results in what?
10. Spherical Jacobian factors
The spherical volume element has rho squared sin phi. As a number, how many factors? Answer with a number.
11. Cone phi limit
The cone phi = pi/6 has the ice cream cone's phi ranging from 0 to pi/6. As a fraction of pi, what is the upper limit? Answer like pi/6.
12. Cylinder radius limit
The cylinder x squared + y squared = 4 has cylindrical r from 0 to what number? Answer with a number.
13. Ice cream cone description
The ice cream cone solid, a cone topped by a spherical cap, in spherical coordinates has phi ranging 0 to the cone's half-angle and rho ranging 0 to what?
14. Choice signal
Spherical coordinates win when the solid involves spheres or cones centered where? Answer with one word.