c Amy Austin, November 28, 2016

Section 14.5 Curl and Divergence In this section, we define two operations on vector fields. These operations are called Divergence and Curl, which are characteristics of how fluid/flow is behaving in a small neighborhood around a given point. Definition: Divergence is a measurement of how much fluid/flow enters the neighborhood around a point P compared to how much fluid/flow exits the neighborhood around P . If more fluid/flow enters the neighborhood around P than leaves the neighborhood around P , the divergence will be negative(gaining fluid/flow in that neighborhood).

If the same amount of fluid/flow enters the neighborhood around P as leaving it, the divergence will be zero.

If more fluid/flow leaves the neighborhood around P than enters the neighborhood around P , the divergence will be positive (losing fluid in that neighborhood). In this case, we say the vector field is divergent at the point P .

Example 1: Consider F = −xj. Can we intuitively determine the divergence of F?

∂ ∂ ∂ Definition: The del operator, denoted by ∇, is defined as ∇ = , , . Note: this is not the ∂x ∂y ∂z  same as the gradient!!

Definition: If F = hP, Q, Ri is a a vector field on ℜ3 and the partial derivatives of P , Q, and R all exist, ∂ ∂ ∂ ∂P ∂Q ∂R then the divergence of F is ∇ · F = , , · hP, Q, Ri = + + . ∂x ∂y ∂z  ∂x ∂y ∂z

1 c Amy Austin, November 28, 2016

Definition: If F = hP, Q, Ri is a vector field on ℜ3 and the partial derivatives of P , Q, and R all exist, then the curl of F is the vector field on ℜ3 defined by curl F = ∇× F.

i j k ∂ ∂ ∂ curl F = ∇× F = ∂x ∂y ∂z

PQR

∂R ∂Q ∂P ∂R ∂Q ∂P curl F = − , − , −  ∂y ∂z ∂z ∂x ∂x ∂y 

Note: The curl of F relates to rotation of a fluid at point P in a neighborhood around that point. In other words the curl measures the tendency of the fluid/flow to rotate in a vector field of a neighborhood around that point. If our vector field is rotating around the z axis, the right hand rule tells us that if the fluid is rotating counterclockwise, thus the curl will point directly upward. If the fluid is rotating clockwise, the curl will point directly downward. Think of circulation as being the amount of pushing, twisting, or turning around the point P . We can visualize curl by the paddle wheel illustration shown below.

If the paddle turns, it has rotation, and thus curl. If the paddle doesn’t turn, there is no curl. Paddle turning means the water is pushing harder on one side than the other, making it twist. The larger the difference, the more forceful the twist, and hence the larger the curl. A turning paddle wheel indicates the field is uneven. If the field were even on all sides equally, the paddle wouldn’t turn at all and the curl would be zero. If that is the case, we call the field irrotational at that point.

2 c Amy Austin, November 28, 2016

Example 2: Find the divergence of F(x,y,z)= x2y,yz2,zx2 at the point (1, −1, 1), (1, 1, 1), and (1, −5, 1). Interpret your answer.

Example 3: Find the divergence and curl of F = hyz,xz,xyi.

Example 4: Find the divergence and curl of F = xy,xz,xyz2 at the point (1, −2, 1).

3 c Amy Austin, November 28, 2016

Theorem: If f is a function of three variables that has continuous second-order partial derivatives, then curl (∇f)= 0. This can be proved by showing curl (∇f)= ∇× (∇f)= 0.

Since a conservative vector field is one for which F = ∇f, this theorem tells us that if F is conservative, then curl (∇f)= curl (F)= 0. Theorem: If F is a vector field defined on all of ℜ3 whose component functions have continuous partial derivatives and curl F = 0, then F is a conservative vector field. This gives us a way to determine whether a vector function on ℜ3 is conservative.

For recall purposes, the ‘conservative test’ on ℜ2: Let F = hP,Qi be a vector field on an open simply- ∂Q ∂P connected region D. Suppose that P and Q have continuous first-order derivatives and = through- dx dy out D. Then F is conservative.

Example 5: Determine if the vector field is conservative. F = y2z3, 2xyz3, 4xy + z . If it is conservative, find the potential function f.

Theorem: If F = hP, Q, Ri is a vector field on ℜ3 and P , Q, and R have continuous second-order partial derivatives, then div curl F = 0.

Example 6: Is there a vector field G on ℜ3 such that curl G = hyz,xyz,zyi?

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