F(t) = |
┌ sin(t) ┐ └ cos(t) ┘ |
G(t) = |
┌ sin(t) ┐ │ 2t │ └ t ┘ |
F((x, y, z)) = |
┌ 2x – 2y + z ┐ │ 3x + y – z │ │ x + 5y │ └ 6x – 3y + 3z ┘ |
∂ φ(x,y,z) ---------- ∂ x |
, | ∂ φ(x,y,z) ---------- ∂ y |
, | ∂ φ(x,y,z) ---------- ∂ z |
∇ φ | = | ∂ φ(x,y,z) ---------- ∂ x |
ê_{x} | + | ∂ φ(x,y,z) ---------- ∂ y |
ê_{y} | + | ∂ φ(x,y,z) ---------- ∂ z |
ê_{z} | (equation 1) |
∇ φ | = |
┌ ∂/∂x φ(x,y,z) ┐ │ ∂/∂y φ(x,y,z) │ └ ∂/∂z φ(x,y,z) ┘ |
(equation 2) |
∇ | = |
┌ ∂/∂x ┐ │ ∂/∂y │ └ ∂/∂z ┘ |
(equation 3) |
F | = |
┌ F_{x}(x,y,z) ┐ │ F_{y}(x,y,z) │ └ F_{z}(x,y,z) ┘ |
∇ · F | = |
┌ ∂/∂x ┐ │ ∂/∂y │ └ ∂/∂z ┘ |
· |
┌ F_{x}(x,y,z) ┐ │ F_{y}(x,y,z) │ └ F_{z}(x,y,z) ┘ |
= | ∂/∂x F_{x}(x,y,z) + ∂/∂y F_{y}(x,y,z) + ∂/∂z F_{z}(x,y,z) | (equation 4) |
A = |
┌ a_{1} ┐ │ a_{2} │ └ a_{3} ┘ |
B = |
┌ b_{1} ┐ │ b_{2} │ └ b_{3} ┘ |
A · B = |
a_{1}b_{1}+a_{2}b_{2}+a_{3}b_{3} |