Lecture 25 (Nov. 14)
Curl and Divergence (reading: 16.5)
Let F = P î + Qĵ + Rk̂ be a vector field on R3 .
Definition: the curl of F is the vector field
curlF = r ⇥ F =
✓
@R
@y
@Q
@z
◆
î +
✓
@P
@z
@R
@x
◆
ĵ +
Definition: the divergence of F is the (scalar) function
divF = r · F =
@P
@Q @R
+
+
.
@x
@y
@z
Remark on notation:
Example: find the divergence and curl of F = hxy, yz, xzi.
16
✓
@Q
@x
@P
@y
◆
k̂.
Theorem: if f (x, y, z) has continuous second partials, then
curl(rf ) = 0.
Proof:
Remark: this theorem says: F conservative =) curlF = 0.
Special case: F = hP (x, y), Q(x, y), 0i (a 2D vector field “in disguise”):
Converse result:
Theorem: if F is a vector field on R3 with continuous partials, and curlF = 0, then
F is conservative.
Proof: later.
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Example: show F = 2xy î + (x2 + 2yz)ĵ + y 2 k̂ is conservative, and find a potential
function for F.
Theorem: suppose F = P î + Qĵ + Rk̂ has continuous second partials. Then
r · (r ⇥ F) = 0.
Proof:
Example: show F = hxy 2 , yz 2 , x2 zi is not a curl.
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