Description
Return signed and unsigned local area factors.
Integration depends on the domain, orientation, coordinate change and endpoint behavior. A boundary contribution, Jacobian factor or power-tail test is a local part of an integral calculation; convergence, a valid change of variables and the complete integral require their own hypotheses.
Return signed and unsigned local area factors.
When to use this tool
Use Calculus: Differential Area Local Scale when evaluating the stated integral contribution, orientation, Jacobian or convergence condition.
How to use the tool and how it works
- Enter jacobian. Use the units and conventions below.
- Run the tool. Correct any format, range, or cross-field input error before interpreting the output.
- Read result. Expand result to inspect the returned fields.
Return signed and unsigned local area factors. Require a 2x2 Jacobian; signedLocalAreaFactor = a*d-b*c, absoluteLocalAreaFactor = abs(determinant).1
| Input field | Meaning and units | Accepted input |
|---|---|---|
| jacobian | Jacobian | Required. Array of array of finite number. Implementation limit: 128 items. |
| Output field | Interpretation | Returned shape |
|---|---|---|
| result | Returned calculated value or nested object under the stated output convention. | Required. Object. |
| result.signedLocalAreaFactor | Signed determinant; zero is returned and does not establish a valid invertible coordinate change. | Calculated result member: number. |
| result.absoluteLocalAreaFactor | Absolute Local Area Factor | Calculated result member: number. |
{
"input": {
"jacobian": [
[
2,
1
],
[
0,
3
]
]
},
"output": {
"result": {
"signedLocalAreaFactor": 6,
"absoluteLocalAreaFactor": 6
}
}
}Limitations and assumptions
- A zero determinant is returned without rejection; no mapping, invertibility neighborhood or change-of-variables hypotheses are verified.
- Outputs are nested inside result. Inputs must satisfy both the field types and the cross-field conditions above. Invalid input causes an error; a successful false flag or documented placeholder is a different result.
Alternative or Complementary methods
Specify and analyze the complete integrand and domain, including endpoint singularities, coordinate invertibility and any omitted boundary or interior terms.
References
Similar or alternative tools
- Logarithmic Coordinate Area Jacobian
Return the area factor from logarithmic coordinates to positive coordinates.
- Graph Surface Area Integrand
Return the graph-surface area integrand.