Calculus Volume 3 · 免费预览
曲面积分
曲面积分在曲面上累积标量，累积量为叉积模的平方根；或者累积场的通量，即法向分量的积分。曲面的参数化是工作工具。
1. Parametrized surfaces and the area element
Parametrized surfaces and the area element
2. Flux integrals
Flux integrals
3. Orientability and the theorems' surfaces
Orientability and the theorems' surfaces
4. Surface integrals of scalars
Surface integrals of scalars
5. Surface area element
The scalar surface integral's area element dS equals which expression?
6. Flux definition
The flux of F across the oriented surface S is which integral?
7. Orientation sign
Reversing the surface's orientation changes the flux integral's sign by what factor?
8. Non-orientable surface
Which surface is the standard example of non-orientability, no consistent continuous normal?
9. Sphere flux uniform field
The constant field F = (0, 0, 1) through the closed sphere: what is the total flux?
10. Sphere area element
The sphere's area element is a squared sin phi d theta d phi. As a power, what is the a's exponent? Answer with a number.
11. Flux constant normal
For a field parallel to the constant normal, the flux is the field's magnitude times which quantity? Answer like the area.
12. Tangent cross product
The surface normal is the cross product of the two tangent vectors. As a two-word phrase, what are they? Answer like tangent plane.
13. Graphs as parametrizations
The graph z = g(x, y) parametrizes as r = (x, y, g(x, y)). What is the magnitude |r_x cross r_y| for this parametrization?
14. Outward convention
Closed surfaces in the divergence theorem use which orientation? Answer with one word.