Description
Return Euler homogeneity residual at supplied values.
Differential-calculus formulas relate supplied local values and derivatives under explicit coordinate and unit conventions. A tangent term, curvature expression or directional derivative assumes the required differentiability and regularity; substituting numerical derivatives does not verify those hypotheses for an actual function.
Return Euler homogeneity residual at supplied values.
When to use this tool
Use Euler Homogeneity Local Residual when evaluating the stated local differential-calculus identity from supplied values.
How to use the tool and how it works
- Enter point, gradient, degree, value. Follow any array-alignment requirements below. Use the units and conventions below.
- Run the tool. Correct any format, range, or cross-field input error before interpreting the output.
- Read residual. Use the output convention below.
Return Euler homogeneity residual at supplied values. Require aligned point/gradient; residual = dot(point,gradient)-degree*value.1
| Input field | Meaning and units | Accepted input |
|---|---|---|
| point | Point | Required. Array of finite number; maximum items 256. |
| gradient | Gradient | Required. Array of finite number; maximum items 256. |
| degree | Degree | Required. Finite number. |
| value | Value | Required. Finite number. |
| Output field | Interpretation | Returned shape |
|---|---|---|
| residual | Signed difference in the direction stated above, with matching units. | Required. Finite number. |
{
"input": {
"point": [
2,
3
],
"gradient": [
4,
6
],
"degree": 2,
"value": 13
},
"output": {
"residual": 0
}
}Limitations and assumptions
- Zero residual at one supplied point does not prove the function is homogeneous or its derivatives correct.
- 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.
- 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. Numeric signed-zero results are exported as zero.
Alternative or Complementary methods
Differentiate or validate the complete function and its domain first; use suitable symbolic or numerical methods and check regularity and coordinate conventions for the intended interpretation.
References
Similar or alternative tools
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