As background, I am in Vector Calculus. My question is giving me some "short cuts" to use to help with evaluating flux integrals. The hint says "If F (Vector Field) is tangent at every pointof S, then the integral of S of F dot dA =0. What does this mean exactly? How can I know if this vector field is tangent?
The second hint is "If F is perpendicular at every point of S and has a constant magnitude on S, then the integral of S of F dot dA = +/- ||F|| dot Area of the Surface (choose the positive sign if F points in the same direction s the orientation of S, Choose negative if F points in the direction opposite the orientation of S)
I am having trouble understand what these hints mean and when/ how to use them.
The Flux integral I have to solve is: F=e^(y^2+z^2)i through the disk of radius 2 in the yz-plane, centered at the origin and oriented in the positive x direction
Presumably, you learned in your class about the normal vector to a surface at a point. Saying the vector field is "tangent to the surface" means that the vector field is perpendicular to the normal vector of the surface. Geometrically/visually, this means the vector points in a direction within the tangent plane at that point.
So if $F$ is the vector field, and $dA$ is the normal to the surface, then $F \cdot dA = 0$, because they are perpendicular. If you integrate zero, you get zero.
The second hint is similar. Saying the vector field is "perpendicular to the surface" means that it points in the same (or opposite) direction of the normal vector everywhere. I guess the notation you are using probably means $dA$ is the unit normal vector. Then $F \cdot dA = \pm |F|$. Remember that if two vectors $v$ and $w$ are parallel, then $|v \cdot w| = |v| |w|$.
So if $|F|$ is the same length everywhere, then $F \cdot dA = \pm |F|$ is a constant function on the surface (it's the same at every point). Integrating a constant function is that constant times the area (this is just a basic principle about double integrals).