Surface Integrals

Parametrized surfaces and the area element

Flux integrals

Orientability and the theorems' surfaces

Surface integrals of scalars

The scalar surface integral's area element dS equals which expression?

The flux of F across the oriented surface S is which integral?

Reversing the surface's orientation changes the flux integral's sign by what factor?

Which surface is the standard example of non-orientability, no consistent continuous normal?

The constant field F = (0, 0, 1) through the closed sphere: what is the total flux?

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.

For a field parallel to the constant normal, the flux is the field's magnitude times which quantity? Answer like the area.

The surface normal is the cross product of the two tangent vectors. As a two-word phrase, what are they? Answer like tangent plane.

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?

Closed surfaces in the divergence theorem use which orientation? Answer with one word.