Calculus Volume 3 · 免费预览
二次曲面
二次曲面是三元二次方程的图形，包括椭球面、单叶双曲面与双叶双曲面、椭圆抛物面和双曲抛物面，它们可按截面和符号分类。
1. The quadric family
The quadric family
2. Paraboloids and cones
Paraboloids and cones
3. Cross sections and standard form
Cross sections and standard form
4. Cylinders as degenerate quadrics
Cylinders as degenerate quadrics
5. Ellipsoid traces
Cross sections of the ellipsoid x^2/a^2 + y^2/b^2 + z^2/c^2 = 1 parallel to the xy plane are which curves?
6. One negative term
The surface x^2/a^2 + y^2/b^2 - z^2/c^2 = 1, with one negative term, is which quadric?
7. Saddle identification
The hyperbolic paraboloid z = x^2/a^2 - y^2/b^2 is commonly called what?
8. Completing the square purpose
Completing the square on a general quadratic brings a quadric's equation to what?
9. Cooling tower shape
The hyperboloid of one sheet is used for cooling towers because of which property?
10. Ellipsoid semi-axis count
The ellipsoid has three semi-axes a, b, and which third letter? Answer with one lowercase letter.
11. Cone degeneracy
The cone is the degenerate quadric separating the hyperboloids, its equation equating z squared to a sum. As a two-word name, what is it called? Answer like double cone.
12. Saddle minimax
The saddle curves upward along one axis and downward along the other, the landscape of which optimization concept? Answer with one word.
13. Sphere as special ellipsoid
The sphere x^2 + y^2 + z^2 = r^2 is the special case of the ellipsoid with what?
14. Negative term count two sheets
The hyperboloid of two sheets has how many negative squared terms? Answer with a number.