Differentiation of vectors
Def: vector
Differentiation of compose vector expression
Let
Then,
Differentiation of a vector
We use a small change
For example, for velocity
,
Intergration of a vector
- If
, then
Vector fuctions of several argument
Chain rule: If
, then Special case:
Differential:
Scalar and vector field
Focus on
- Some region
- 3-dimension only
continous and diffentiable - line, surface, volume
Vector Operators
Gradient
Def:
Def:
Remark:
Infinitesimal change in
from to is where
is the position in the field. If
represents a curve in space, then total deraviative of respected to along the curve is The change rate of
w.r.t (was respected to) distance along is given by is called directional dedrivative
lies on the direction of the fast icrease in .
Chain rule:
Divergence
Def:
Remark:
Def: Laplacian operator
for a scalar field
Curl
Def:
Remark:
Remark: Vector operator formaula:
Combination of grad. div. curl
All posibilities:
Remark:
Cylindrical and spherical polar coordiates
Cylindrical polar coordinates
Def:
The position vector:
The basis vectors:
where
therefore,
The vector operators:
the volume
Spherical polar coordinates
Def:
the basis vector:
General displacement vector
the volume of infinitesimal parallelpiped
The vector operators:
Remark: for simplify the calculation,
General curvilinear coordinates
Def: the position of a point P having Cartesian coordinates x,y,z may be expressed in terms of the three curvilinear coordinates u1,u2,u3,where
and similarly
If
where
infinitesimal vector displacement:
The linear operators are given by
Last Updated on 3 years by Yichen Liu