Jacobian Surface Calculator

Surface Input Form

Enter a parametric surface r(u,v) = (x(u,v), y(u,v), z(u,v)). Supported functions include sin, cos, tan, sqrt, exp, log, pow, min, and max.

x(u,v)

y(u,v)

z(u,v)

Evaluation point u

Evaluation point v

Differentiation step h

Area bound u minimum

Area bound u maximum

u subdivisions

Area bound v minimum

Area bound v maximum

v subdivisions

Example Data Table

These sample parameterizations help you test curvature, area scaling, and normal behavior across common surfaces.

Surface	x(u,v)	y(u,v)	z(u,v)	Suggested bounds	Known Jacobian pattern
Paraboloid	u cos(v)	u sin(v)	u²	u: 0 to 2, v: 0 to π/2	u √(1 + 4u²)
Sphere	sin(u) cos(v)	sin(u) sin(v)	cos(u)	u: 0 to π, v: 0 to 2π	sin(u)
Cylinder	cos(v)	sin(v)	u	u: 0 to 3, v: 0 to 2π	1
Torus	(2 + cos(u)) cos(v)	(2 + cos(u)) sin(v)	sin(u)	u: 0 to 2π, v: 0 to 2π	2 + cos(u)

Formula Used

Surface map: r(u,v) = (x(u,v), y(u,v), z(u,v))

Numerical partials: r_u ≈ [r(u+h,v) − r(u−h,v)] / (2h)

Numerical partials: r_v ≈ [r(u,v+h) − r(u,v−h)] / (2h)

Surface Jacobian: J_s = |r_u × r_v|

Metric terms: E = r_u·r_u, F = r_u·r_v, G = r_v·r_v

Area estimate: Area ≈ ΣΣ J_s(u_i,v_j) Δu Δv

The calculator uses central differences for stable local derivatives, then converts those derivatives into tangent vectors, normals, metric coefficients, and a midpoint-rule surface area estimate over the selected parameter rectangle.

How to Use This Calculator

Enter the three coordinate functions describing your surface in terms of u and v.
Choose the evaluation point where you want local tangent, normal, and Jacobian information.
Set a small differentiation step. Smaller values improve precision until rounding noise becomes noticeable.
Enter parameter bounds and subdivisions if you want a numerical surface area estimate.
Press the calculate button and review the result panel above the form.
Export the result table as CSV or save the panel as PDF for notes, classes, or reports.

FAQs

1. What does the surface Jacobian measure?

It measures local area scaling for a parametric surface. A larger value means a small rectangle in the parameter plane stretches into a larger patch on the surface.

2. Why are tangent vectors shown?

The two tangent vectors define the local surface plane. Their cross product gives the normal direction and its magnitude gives the Jacobian needed for area calculations.

3. Why does the tool ask for a step size?

The calculator uses numerical differentiation. The step size controls how the partial derivatives are approximated, so very large or extremely tiny values can reduce accuracy.

4. What are E, F, and G?

They are coefficients of the first fundamental form. Together, they describe local stretching, angle distortion, and the squared area factor through E G minus F squared.

5. Can I estimate total surface area here?

Yes. Provide parameter bounds and subdivision counts. The tool applies a midpoint double sum of the Jacobian over that rectangle to estimate total area numerically.

6. Which functions can I enter?

You can use u, v, numbers, operators, parentheses, commas, and common functions such as sin, cos, tan, sqrt, exp, log, pow, min, and max.

7. What causes a zero Jacobian?

A zero Jacobian means the tangent vectors become dependent or collapse. Locally, the parameterization loses area and may indicate a singular point or degenerate patch.

8. Is this exact or approximate?

The results are numerical approximations. They are usually very good for smooth surfaces and sensible step sizes, but symbolic algebra is not being performed here.