Scalar Fields
 Scalar Functions = f(x), f(x,y), f(x,y,z)
 Think of x,y,z as space (3D)
 Temperature T(x,y,z), is a scalar field
 The gradient of a scalar field, is a vector field
Vector Fields
Vector = magnitude and direction

Velocity, Acceleration, Force

When vectors vary in space, you have a vector field.
 Each component of a vector field can vary in space, and contains a scalar field.
Important Properties of Vector Fields

Every field has the following properties that can be observed/studied/

These properties can tell us how a field varies in space

Gradience

Divergence

Curl

Divergence is result of dot product, and measures 3 of the three derivatives available
 Divergence measures the “Flow” of a point. If there’s stuff going in, or stuff going out of a point

Curl is the result of cross product, and measures the other 6 of the derivatives available
 Curl measures the “Rotation or swirl” around a point
 If in 2D for curl, you must only keep the last term.
Special Cases
a) Conservative When v =∇f (f is the potential function for v)
Some examples of where this can occur:

Electrical Force Electrical Potential

Magnetic Force Magnetic Potential

Gravitational Force Gravatational Potential
This case occurs when the curl = 0