Solve field behavior across curved surfaces quickly. Evaluate points, tangents, normals, projections, and orientation choices. Download organized outputs for classwork, checking, reporting, and review.
| Surface | Field | (u, v) | Point | Normal Vector | Raw Flux Density |
|---|---|---|---|---|---|
| r(u,v) = (u, v, uv) | F = (x, y, z) | (1, 2) | (1, 2, 2) | (-2, -1, 1) | -2.000000 |
| r(u,v) = (u, v, u+v) | F = (0, 0, 1) | (1, 2) | (1, 2, 3) | (-1, -1, 1) | 1.000000 |
For a parametric surface r(u,v) = (x(u,v), y(u,v), z(u,v)), the calculator first evaluates the point on the surface.
It then computes the tangent vectors:
ru = ∂r/∂u and rv = ∂r/∂v
The oriented normal vector is:
n = ru × rv
The unit normal is:
n̂ = n / |n|
If the vector field is F(x,y,z) = (P, Q, R), then the field value at the surface point is found by substitution.
Local raw flux density is:
F · n
Normal component along the unit normal is:
F · n̂
The tangential part is:
FT = F - (F · n̂)n̂
A vector field on a surface shows how a directed quantity behaves at each point. This calculator studies that behavior on a parametric surface. It finds the surface point first. Then it computes tangent vectors, the normal vector, and the unit normal. It also evaluates the vector field at the same location. These results help you inspect direction, strength, and local flow through the surface. That makes the tool useful for calculus, geometry, physics, and engineering revision.
The surface normal is central in many multivariable problems. It tells you which direction is perpendicular to the surface. Once the normal is known, you can separate the field into normal and tangential parts. The normal part is important in flux questions. The tangential part is important in sliding or surface-parallel behavior. Reversing the orientation changes the sign of the normal-based result. That is why this calculator also includes a reverse normal option.
The calculator returns more than one answer. It reports the point on the surface, the tangent vectors, the raw normal vector, and the unit normal. It also shows the field value, field magnitude, raw flux density, normal projection, tangential vector, tangential magnitude, and the angle with the unit normal. These outputs help you verify homework steps. They also help you check symbolic work with a quick numerical test at a chosen parameter pair.
Use clear expressions and write multiplication with an asterisk. Trigonometric entries should use radians. Good test cases include planes, cylinders, spheres, and graph surfaces written in parametric form. If the normal becomes zero, choose a different point. That usually means the parameterization is singular there. For classroom work, compare the calculator result with your manual cross product and dot product steps. This builds confidence and improves accuracy during practice.
It computes the surface point, tangent vectors, normal vector, unit normal, field value, raw flux density, normal component, tangential component, and angle with the unit normal.
Enter a parametric surface using x(u,v), y(u,v), and z(u,v). This format works well for planes, cylinders, spheres, saddles, and many standard calculus examples.
Enter the vector field as three scalar components P(x,y,z), Q(x,y,z), and R(x,y,z). The calculator substitutes the surface point into those expressions.
Raw flux density is F · (ru × rv). It measures oriented flow through the parameterized surface patch before dividing by the normal magnitude.
The sign changes when the surface orientation changes. Reversing the normal flips normal-based quantities like flux density and the signed normal component.
No. Trigonometric functions use radians. Convert degree values before entering them if your problem statement is written in degrees.
That means the tangent vectors became dependent at the chosen point. In practice, the parameterization is singular there, so choose another parameter pair.
Yes. It is useful for checking cross products, dot products, orientation choices, and the split between normal and tangential field behavior.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.