Laplacian (Δ)
The divergence of a conservative field is equal to the laplacian of the potential! It is marked as ΔF, or ∇^{2}
Incompressible Fields
 The Curl of a conservative field will always be ZERO
 The Divergence of a incompressible field will always be ZERO
In 2D, an incompressible field is associated with stream functions (similar to potential functions)
 Incompressible fields have different markings!
compared to
$v=(v_{x},v_{y})$ Stream functions do NOT exist in 3D, only 2D
Conservative and Incompressible Fields
If conservative and Incompressible, the field will satisfy Laplace’s Equation
Fields and Curves
Recall that r(t)..
$r(t)=(x,(t),y(t),z(t))$Over a range of t, we will generate a curve.
 We have seen this before!
Recall that the unit tangent vector T =
$T^=∣∣dtdr ∣∣dtdr =∣∣v∣∣v $ Recall our smooth curves!
Convention
 We travel counter clockwise around the curve.
We can use these curves for our fields!
Scalar Fields f(x,y) or f(x,y,z) Curve r(t) = (x(t),y(t),z(t))
What is the scalar function along the curve?
Vector fields f(x,y) or f(x,y,z) Curve r(t) = (x(t),y(t),z(t))
What is f on curve?

Plug in x(t), y(t) and z(t) from curve into f → f(t)

We can break f into two components!
 The Tangential Component
 The Normal Component
 One thing to note about N, is that it is not the same as it used to be!
 N now will face the other direction than it did previously
 Before, N used to face inwards during the curve.
 Now, it faces outwards.
Now,
$N^=T^×K^$N is the normal T is the tangent K is the Khat from (i, j, k)