Description
Return the area factor from logarithmic coordinates to positive coordinates.
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 the area factor from logarithmic coordinates to positive coordinates.
When to use this tool
Use Logarithmic Coordinate Area Jacobian when evaluating the stated local differential-calculus identity from supplied values.
How to use the tool and how it works
- Enter x, y, base. Use the units and conventions below.
- Run the tool. Correct any format, range, or cross-field input error before interpreting the output.
- Read jacobian. Use the output convention below.
Return the area factor from logarithmic coordinates to positive coordinates. Require x,y > 0 and base > 0 except one; jacobian = x*y*ln(base)².1
| Input field | Meaning and units | Accepted input |
|---|---|---|
| x | X | Required. Finite number; strictly above 0. |
| y | Y | Required. Finite number; strictly above 0. |
| base | Base | Required. Finite number; strictly above 0. |
| Output field | Interpretation | Returned shape |
|---|---|---|
| jacobian | dx dy / d(log_b x) d(log_b y) = x*y*ln(base)^2, not its reciprocal. | Required. Finite number. |
{
"input": {
"x": 2,
"y": 3,
"base": 2.718281828459045
},
"output": {
"jacobian": 6
}
}Limitations and assumptions
- This is dx*dy per d(log_base x)*d(log_base y), not its reciprocal; no region or integral is transformed.
- 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
- Logarithmic Derivative Residue Term
Return a logarithmic-derivative residue from zero/pole orders.
- Logarithmic Time Derivative Factor
Convert a time derivative to a natural-log-time derivative.