Description
Return a second derivative for a supplied implicit planar curve.
An implicit relation defines local derivatives only where its smoothness and nonzero-partial hypotheses apply. Gradient and Hessian components supplied as numbers do not establish that a point lies on the level set or that a regular implicit function exists there.
Return a second derivative for a supplied implicit planar curve.
When to use this tool
Use Implicit Second Partial Diagonal Term when evaluating the stated implicit-derivative or level-set term from supplied partial derivatives.
How to use the tool and how it works
- Enter fx, fy, fxx, fxy, fyy. Use the units and conventions below.
- Run the tool. Correct any format, range, or cross-field input error before interpreting the output.
- Read secondDerivative. Use the output convention below.
Return a second derivative for a supplied implicit planar curve. Require fy != 0; y1 = -fx/fy, secondDerivative = -(fxx+2*fxy*y1+fyy*y1²)/fy.1
| Input field | Meaning and units | Accepted input |
|---|---|---|
| fx | Fx | Required. Finite number. |
| fy | Fy | Required. Finite number. |
| fxx | Fxx | Required. Finite number. |
| fxy | Fxy | Required. Finite number. |
| fyy | Fyy | Required. Finite number. |
| Output field | Interpretation | Returned shape |
|---|---|---|
| secondDerivative | Second Derivative | Required. Finite number. |
{
"input": {
"fx": 2,
"fy": 4,
"fxx": 6,
"fxy": 1,
"fyy": 2
},
"output": {
"secondDerivative": -1.375
}
}Limitations and assumptions
- All partials must come from the same sufficiently differentiable level-set point; existence and mixed-partial conventions are supplied assumptions.
- 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
Check the original level-set equation, differentiability and the relevant implicit-function hypotheses before interpreting these local derivative formulas.
References
Similar or alternative tools
- Implicit Three Variable Partial Ratio
Return local partials for a supplied implicit graph z(x,y).
- Implicit Levelset Gradient Normalizer
Normalize a supplied nonzero level-set gradient.