Vector Calculus (Chapter 13) 9

Vector Calculus Chapter 13 13.1 13.2-13.3 13.4 Scalar Fields, 2D Vector Fields Gradient Vector Fields Line Integrals Green’s Theorem

F(x,y)=<P(x,y),Q(x,y)>

Scalar Fields and Vector Fields The simplest possible physical field is the scalar field. It represents a function depending on the position in space. A scalar field is characterized at each point in space by a single number. Examples of scalar fields temperature, gravitational potential, electrostatic potential (voltage)

Scalar Fields Visualization of z=V(x,y) Scalar potential function for a dipole V(x,y)

Maple commands

Scalar Fields and Equipotential Lines The level curves or contours of the function z=V(x,y) are the equipotential lines of the scalar potential field V(x,y)

The Gradient defines a Vector Field (the force field)

Arrow Diagram for Vector Field

Direction Field (magnitude=1)

Equipotential surfaces are orthogonal to the electric force field Notice the force field is directed towards places where the potential V is lower, e.g., where the charge is negative - at (0.25,0). But mathematically, the gradient points in the opposite direction (greatest ascent) which is why f=-V and F=grad(f)=grad(-V)

2D vector field visualization of the flow field past an air foil using arrows