Tensor_8 Some Basic Concepts and Notations

Differentiating displacement to a scalar such as time gets vector – velocity, this velocity vector can be further differentiated by the same scalar time to get acceleration etc.; differentiating a scalar such as temperature to displacement in a room space gets gradient; both are vector but different in this sense. Differentiating a vector to another vector can also get a new vector.
Basis vector is defined as the derivatives of position vector R to the directional/axis vector (u or v, x, or y or z). If it’s normalized to unit e vector then it’s length = 1. Why?

e1 = partial R/partial x axis = scalar value for component * unit e1 vector; so the unit e1 vector = R'(s) projected on x axis divided by the length of it, because this scalar value for component is the length, it’s also equals R’ dot e1 = |R’||e1|costheta.

Often case we see R(Z1(s)) and R(Z2(s)).

It’s intuitive from above visual that the first derivative it a tangent line of curve S formed by position vector R. What is the visual intuition of the second derivative R(s)”. It is exactly the acceleration , complex laid out by eigenchris as

It’s mind blowing that the first derivative of position vector R to lambda is tangent line; dR to du or dv is the basis vector, the second derivative can explode as above equation. The dR to du twice can be interrelated as acceleration, perpendicular to dR/du, to make of the dR to du and then to dv, as shown below, it leads to the curvature. The Christoffel symbol is said to measure the change rate of basis vector when u change…

3. Differentiate these concepts: g metric tensor, kronecker delta, identity matrix, Jacobean matrix from simple to difficult

kronecker delta is identity matrix, the former is a good notation to express the latter

Jacobean matrix

Jf Jb = identity matrix.

Cartesian coordinates, metric tensor is identity, however,

note

in polar coordinates, different story, for example the below is the metric tensor of polar system and of Einstein’s GR equation.

In the end, it’s worthy to grasp the definition of basis vector, using polar system as an example.

Note the basis vector is defined as derivative of positional vector to coordinate, or rate of positional vector length (?) change per coordinate in polar system, er is easier in each place the value is same; while etheta is different when r change.