Description
Calculate one Jacobian entry of vector normalization with respect to an input coordinate.
A vector is an ordered set of components in a chosen basis. Its norm measures magnitude, while dot products, projections, and coordinate transformations describe different relationships. Component order, angle units, normalization, and numerical tolerances determine the meaning of a computed result.
Calculate one Jacobian entry of vector normalization with respect to an input coordinate.
When to use this tool
Use Vector Normalization Coordinate Derivative when you need the particular coordinate operation, derivative, residual, or domain check under a known basis convention.
How to use the tool and how it works
- Enter vector, row, column. Keep parallel arrays aligned. Use the units and conventions below.
- Run the tool. Correct any format, range, or cross-field input error before interpreting the output.
- Read derivative. Use the output convention below.
Require valid zero-based row and column indices and a positive vector norm. Return one derivative entry of vector normalization: (delta(row,column)-(vector[row]/norm)*(vector[column]/norm))/norm.1
| Input field | Meaning and units | Accepted input |
|---|---|---|
| vector | Vector | Required. Array of Finite number.; maximum items 256. |
| row | Zero-based output-component index. | Required. Safe integer; minimum 0. |
| column | Zero-based input-coordinate index. | Required. Safe integer; minimum 0. |
| Output field | Interpretation | Returned shape |
|---|---|---|
| derivative | Derivative | Required. Finite number. |
{
"input": {
"vector": [
3,
4
],
"row": 0,
"column": 1
},
"output": {
"derivative": -0.096
}
}Limitations and assumptions
- This is a Jacobian entry with respect to an input coordinate, not a normalized vector or finite-difference approximation. The derivative is undefined at the zero vector and can be large near it.
- The field tables state structural bounds; the mechanics also specify cross-field constraints. Malformed inputs and unsupported domains cause errors. A successful false flag, empty branch value, or zero placeholder has the meaning stated above and is distinct from an input error.
- Numeric calculations use binary floating point unless exact decimal-integer arithmetic is explicitly stated. No hidden measurement tolerance or decimal rounding is added; nonfinite or out-of-contract results are rejected. Signed zero is exported as zero.
Alternative or Complementary methods
Choose a complete vector or matrix operation when a single component or residual is insufficient. For sensitive near-zero or near-singular cases, inspect numerical conditioning and use an appropriate precision and tolerance.
References
Similar or alternative tools
- Vector Cross Product One Input Coordinate Derivative
Differentiate first cross second with respect to one coordinate of the first vector, holding the second fixed.
- Vector 3d Coordinate Cyclic Permutation
Cyclically reorder three coordinates by an integer shift modulo three.