MATH 251 – LECTURE 28
JENS FORSGÅRD
http://www.math.tamu.edu/~jensf/
This week: 14.3–4
webAssign: 14.3–4, due 4/11 11:55 p.m.
Next week: 14.5–6
webAssign: 14.5–6, opens 4/11 12 a.m.
Help Sessions:
Sun–Thu 6–8 p.m. in BLOC 149
Office Hours:
BLOC 641C
M 12:30–2:30 p.m.
W 2–3 p.m.
or by appointment.
∇-calculus
In R2: recall that ∇f = hfx0 , fy0 i.
∇-calculus
In R3: recall that ∇f = hfx0 , fy0 , fz0 i.
∇-calculus
Definition 1. Let F be a three dimensional vector field. Then, the curl of F , denoted curl(F ), is defined by
curl(F ) = ∇ × F.
Theorem 2. Let F be a three-dimensional vector field. Then F is conservative if and only if
curl(F ) = h0, 0, 0i.
∇-calculus
Exercise 3. Let F = hy, −x, 0i. Compute curl(F ).
∇-calculus
Exercise 4. Let F = h0, −x2, 0i. Compute curl(F ).
∇-calculus
Definition 5. The divergence of a three dimensional vector field F is
div(F ) = ∇ · F.
Exercise 6. Compute the divergence of the vector field F = hx, y, zi.
∇-calculus
Exercise 7. Compute the divergence of the vector field F = h−x, −y, −zi.
∇-calculus
Exercise 8. Let F be a vector field. Compute div(curl(F )).
∇-calculus
Exercise 9. Let f (x, y, z) be a function. Compute div(∇f ).