1. Vector Calculus (Chapter 13)9

2. 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

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

4. 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)

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

7. Maple commands

8. 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)

9. The Gradient defines a Vector Field (the force field)

10. Arrow Diagram for Vector Field

11. Direction Field (magnitude=1)

13. 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)

14. 2D vector field visualization of the flow field past an air foil using arrows