Calculator Form
Example Data Table
| Function | x | y | f(x,y) | ∂f/∂x | ∂f/∂y |
|---|---|---|---|---|---|
| x^2 + x*y + y^2 | 1 | 2 | 7 | 4 | 5 |
| x^2 + x*y + y^2 | 2 | 1 | 7 | 5 | 4 |
| sin(x) + cos(y) | 0 | 0 | 1 | 1 | 0 |
Formula Used
Main evaluation: z = f(x,y)
Partial derivative with respect to x: ∂f/∂x ≈ [f(x+h,y) - f(x-h,y)] / 2h
Partial derivative with respect to y: ∂f/∂y ≈ [f(x,y+h) - f(x,y-h)] / 2h
Gradient magnitude: |∇f| = √[(∂f/∂x)² + (∂f/∂y)²]
Estimated differential change: dz ≈ (∂f/∂x)dx + (∂f/∂y)dy
Tangent plane approximation: z ≈ f(x₀,y₀) + fx(x - x₀) + fy(y - y₀)
Hessian determinant: H = fxxfyy - (fxy)²
This page supports expressions using x and y, standard functions, powers, constants, and numerical derivative checks.
How to Use This Calculator
1. Enter a valid function in terms of x and y.
2. Supply the point where you want the value and derivatives.
3. Set h for numerical differentiation accuracy.
4. Enter dx and dy if you want a differential estimate.
5. Define graph ranges for x and y.
6. Choose grid points to control surface detail.
7. Press Calculate to display the result above the form.
8. Use CSV and PDF buttons to export your output.
Use explicit multiplication, like 2*x or x*y. Use log(x) for base-10 and ln(x) for natural logarithm.
Frequently Asked Questions
1. What does this calculator solve?
It evaluates a function of two variables at a chosen point. It also estimates partial derivatives, gradient size, second derivatives, Hessian determinant, tangent change, and a surface plot.
2. Which functions are supported?
You can use sin, cos, tan, asin, acos, atan, sinh, cosh, tanh, sqrt, abs, exp, ln, log, pow, floor, ceil, round, min, max, pi, and e.
3. Why must I use explicit multiplication?
The expression parser reads operators directly. Writing x*y, 2*x, or (x+y)*y avoids ambiguity and keeps evaluation safer and more reliable.
4. What is the derivative step h?
h is the small change used for numerical differentiation. Smaller values may improve accuracy, but extremely tiny values can increase rounding noise.
5. What does the Hessian determinant tell me?
It helps classify a stationary point. Combined with second derivatives, it can suggest a local minimum, local maximum, saddle point, or an inconclusive result.
6. What is dz in the output?
dz is the estimated change in function value using differentials. It uses partial derivatives with the entered dx and dy values near the selected point.
7. What is included in the CSV export?
The CSV file includes the summary output and the graph grid data. That makes it suitable for review, reuse, plotting, or reporting elsewhere.
8. What is included in the PDF export?
The PDF export includes the main result summary and the grid table used for the plotted surface. It provides a clean printable record.