Functional Differential Geometry
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Why does d/dx appear in the book’s code (e.g. when defining a vector field like e0) before it is ever explicitly defined with a define, and why is it unbound until you run certain coordinate-system definitions?
In the FDG library (scmutils), the symbols d/dx, d/dy, d/dr, d/dθ, etc. are automatically created as literal vector-field basis objects the first time you define coordinates on a manifold using define-coordinates.
Specifically:
; rectangular coordinates (define-coordinates (up x y) R2-rect)
does two things behind the scenes:
- It creates coordinate functions
x,y(orr,theta) that go from points → numbers. - It simultaneously creates the dual basis vector fields named
d/dx,d/dy(ord/dr,d/dθ) that are literal vector fields on that coordinate system.
(for-each (lambda (name) (environment-define env (string->symbol (string-append "d/d" (symbol->string name))) (make-literal-vector-field name coord-sys))) coordinate-names)
These d/d… objects are not defined by a visible define in the user code; they are inserted into the global environment by the macro define-coordinates itself.
— Me@2025-10-12 10:49:20 AM
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2025.12.07 Sunday (c) All rights reserved by ACHK
is just the meaning of the definition of 
. The second factor is needed because the input of 
, not 
.
is just the definition of 
.
is the direction derivative of the function 
at the point 
.
‘s and 
‘s represent the amount and direction of changes respectively.
that gives the altitude at each point on the manifold. Let 
be a vector field on the manifold, perhaps specifying a direction and rate of walking at every point on the manifold.
be a real-valued function, 
be a vector field on the manifold function respectively:
![\begin{aligned} a_i(x)&=\boldsymbol{\omega}(\text{X}_i)(\chi^{-1}(x)) \\ &= (a' \circ \chi') \tilde{\text{X}'}(\text{X}_i) (\chi^{-1}(x)) \\ \\ \tilde{\text{X}}'(\text{X}_i) &= \tilde{\text{X}}(\text{X}_i) \left[ (D(\chi \circ (\chi')^{-1}) \circ \chi' \right]^{-1} \\ \\ a_i(x) &= (a' \circ \chi') \tilde{\text{X}}(\text{X}_i) \left[ D(\chi \circ (\chi')^{-1}) \circ \chi' \right]^{-1} (\chi^{-1}(x)) \\ &= \sum_j (a'_j \circ \chi') \tilde{\text{X}^j}(\text{X}_i) \left[ D(\chi \circ (\chi')^{-1}) \circ \chi' \right]^{-1} (\chi^{-1}(x)) \\ &= (a'_i \circ \chi') \left[ D(\chi \circ (\chi')^{-1}) \circ \chi' \right]^{-1} (\chi^{-1}(x)) \\ \end{aligned}](https://s0.wp.com/latex.php?latex=%5Cbegin%7Baligned%7D++a_i%28x%29%26%3D%5Cboldsymbol%7B%5Comega%7D%28%5Ctext%7BX%7D_i%29%28%5Cchi%5E%7B-1%7D%28x%29%29+%5C%5C++%26%3D+%28a%27+%5Ccirc+%5Cchi%27%29+%5Ctilde%7B%5Ctext%7BX%7D%27%7D%28%5Ctext%7BX%7D_i%29+%28%5Cchi%5E%7B-1%7D%28x%29%29+%5C%5C+%5C%5C++++%5Ctilde%7B%5Ctext%7BX%7D%7D%27%28%5Ctext%7BX%7D_i%29+%26%3D+%5Ctilde%7B%5Ctext%7BX%7D%7D%28%5Ctext%7BX%7D_i%29+%5Cleft%5B+%28D%28%5Cchi+%5Ccirc+%28%5Cchi%27%29%5E%7B-1%7D%29+%5Ccirc+%5Cchi%27+%5Cright%5D%5E%7B-1%7D+%5C%5C+%5C%5C+++++a_i%28x%29++%26%3D+%28a%27+%5Ccirc+%5Cchi%27%29++%5Ctilde%7B%5Ctext%7BX%7D%7D%28%5Ctext%7BX%7D_i%29+%5Cleft%5B+D%28%5Cchi+%5Ccirc+%28%5Cchi%27%29%5E%7B-1%7D%29+%5Ccirc+%5Cchi%27+%5Cright%5D%5E%7B-1%7D++%28%5Cchi%5E%7B-1%7D%28x%29%29+%5C%5C+++%26%3D+%5Csum_j+%28a%27_j+%5Ccirc+%5Cchi%27%29+%5Ctilde%7B%5Ctext%7BX%7D%5Ej%7D%28%5Ctext%7BX%7D_i%29+%5Cleft%5B+D%28%5Cchi+%5Ccirc+%28%5Cchi%27%29%5E%7B-1%7D%29+%5Ccirc+%5Cchi%27+%5Cright%5D%5E%7B-1%7D++%28%5Cchi%5E%7B-1%7D%28x%29%29+%5C%5C+++%26%3D+%28a%27_i+%5Ccirc+%5Cchi%27%29++%5Cleft%5B+D%28%5Cchi+%5Ccirc+%28%5Cchi%27%29%5E%7B-1%7D%29+%5Ccirc+%5Cchi%27+%5Cright%5D%5E%7B-1%7D++%28%5Cchi%5E%7B-1%7D%28x%29%29+%5C%5C+++++%5Cend%7Baligned%7D&bg=ffffff&fg=333333&s=1&c=20201002)
![\begin{aligned} a_i(\chi(\mathbf{m})) &= (a'_i \circ \chi') \left[ D(\chi \circ (\chi')^{-1}) \circ \chi' \right]^{-1} (\mathbf{m}) \\ &= (a'_i (\chi'(\mathbf{m}))) \left[ D(\chi \circ (\chi')^{-1}) (\chi'(\mathbf{m}))) \right]^{-1} \\ \end{aligned}](https://s0.wp.com/latex.php?latex=%5Cbegin%7Baligned%7D++a_i%28%5Cchi%28%5Cmathbf%7Bm%7D%29%29++%26%3D+%28a%27_i+%5Ccirc+%5Cchi%27%29++%5Cleft%5B+D%28%5Cchi+%5Ccirc+%28%5Cchi%27%29%5E%7B-1%7D%29+%5Ccirc+%5Cchi%27+%5Cright%5D%5E%7B-1%7D++%28%5Cmathbf%7Bm%7D%29+%5C%5C+++++%26%3D+%28a%27_i+%28%5Cchi%27%28%5Cmathbf%7Bm%7D%29%29%29++%5Cleft%5B+D%28%5Cchi+%5Ccirc+%28%5Cchi%27%29%5E%7B-1%7D%29+%28%5Cchi%27%28%5Cmathbf%7Bm%7D%29%29%29+%5Cright%5D%5E%7B-1%7D+++%5C%5C+++++%5Cend%7Baligned%7D&bg=ffffff&fg=333333&s=1&c=20201002)
![a(\chi(\mathbf{m})) = a'(\chi'(\mathbf{m})) \left[D(\chi \circ (\chi')^{-1})(\chi'(\mathbf{m}))\right]^{-1}](https://s0.wp.com/latex.php?latex=a%28%5Cchi%28%5Cmathbf%7Bm%7D%29%29+%3D+a%27%28%5Cchi%27%28%5Cmathbf%7Bm%7D%29%29+%5Cleft%5BD%28%5Cchi+%5Ccirc+%28%5Cchi%27%29%5E%7B-1%7D%29%28%5Cchi%27%28%5Cmathbf%7Bm%7D%29%29%5Cright%5D%5E%7B-1%7D&bg=ffffff&fg=333333&s=1&c=20201002)
is a linear combination of coordinate-basis one-form fields:
are smooth functions and 
are the differentials of the coordinates.
, a vector independent of 
, with 
, a vector function of 
is given by:
is the coefficient function for the vector field 
. It provides a scale factor for the component in each coordinate direction.
has a coordinate representation 
: 
and 
.
is the direction derivative of the function 
at the point 
.
, a vector independent of 
, with 
or 
with 
, which is a vector function chosen by you.
with respect to a vector 
at a point (e.g., position) 
may be denoted by any of the following: 
be a differentiable manifold and 
a point of 
is a function defined in a neighborhood of 
(see Exterior derivative), 
(see Covariant derivative), 
(see Lie derivative), or 
(see Tangent space § Definition via derivations), can be defined as follows. ![\displaystyle{\gamma: [-1, 1] \to M}](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle%7B%5Cgamma%3A+%5B-1%2C+1%5D+%5Cto+M%7D&bg=EAEFF3&fg=333333&s=1&c=20201002)
be a differentiable curve with 
and 
. Then the directional derivative is defined by 
, provided 
in 
, one defines the corresponding directional derivative at a point 
by![\displaystyle{ \forall f\in {C^{\infty }}(\mathbb {R} ^{n}):\qquad (D_{v}f)(x):=\left.{\frac {\mathrm {d} }{\mathrm {d} {t}}}[f(x+tv)]\right|_{t=0}=\sum _{i=1}^{n}v^{i}{\frac {\partial f}{\partial x^{i}}}(x).}](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle%7B+%5Cforall+f%5Cin+%7BC%5E%7B%5Cinfty+%7D%7D%28%5Cmathbb+%7BR%7D+%5E%7Bn%7D%29%3A%5Cqquad+%28D_%7Bv%7Df%29%28x%29%3A%3D%5Cleft.%7B%5Cfrac+%7B%5Cmathrm+%7Bd%7D+%7D%7B%5Cmathrm+%7Bd%7D+%7Bt%7D%7D%7D%5Bf%28x%2Btv%29%5D%5Cright%7C_%7Bt%3D0%7D%3D%5Csum+_%7Bi%3D1%7D%5E%7Bn%7Dv%5E%7Bi%7D%7B%5Cfrac+%7B%5Cpartial+f%7D%7B%5Cpartial+x%5E%7Bi%7D%7D%7D%28x%29.%7D&bg=EAEFF3&fg=333333&s=1&c=20201002)
. Furthermore, every derivation at a point in 
is of this form. Hence, there is a one-to-one correspondence between vectors (thought of as tangent vectors at a point) and derivations at a point. 
and
,
is a coordinate function.
maps points in a coordinate patch of a manifold to a coordinate tuple: 
, 
may have a convenient tuple structure.
is a further generalization of 
.
has a coordinate representation 
:
and 
.
is the tuple of coordinate components of a point, 
is the abstract point itself.
inputs an abstract point and outputs its coordinates.
is just the meaning of the definition of 
. The second factor is needed because the input of 
, not 
.
is just the definition of 
.
is a generalization of 
is written as subscript to capture the meaning of “with respect to 
along the direction of the vector 
is
is called directional derivative. 
at 
is independent of 
.
is invariant under change of coordinates …
:
?
.
?
be a point on a manifold, 
be a vector on the manifold, and 
be a real-valued function on the manifold.
.
is the coefficient function for the vector field 
for the coordinates 
(see equation 3.7).
and the points on the sphere are specified both by Riemann coordinates and the traditional colatitude and longitude 
.
Physics Town 
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