Analyze component-wise Laplacians for vector fields quickly. Enter polynomial coefficients, inspect results, and download summaries. Study multivariable behavior with tables, formulas, graphs, and exports.
| Vector Field | Point | Vector Laplacian Formula | Result |
|---|---|---|---|
| <x^3 + y^2 + 2z^2, 2x^2 + y^3 + z, 3z^3 + xy + 4> | (2, 1, 1) | <6x + 6, 6y + 4, 18z> | <18, 10, 18> |
The vector Laplacian acts on each component of a vector field separately.
For a field F = <P, Q, R>, the operator is:
∇²F = <∇²P, ∇²Q, ∇²R>
Each scalar Laplacian is:
∇²P = ∂²P/∂x² + ∂²P/∂y² + ∂²P/∂z²
∇²Q = ∂²Q/∂x² + ∂²Q/∂y² + ∂²Q/∂z²
∇²R = ∂²R/∂x² + ∂²R/∂y² + ∂²R/∂z²
For the polynomial template used here, mixed terms, linear terms, and constants disappear after second differentiation. Cubic terms become linear terms, and squared terms become constants.
So for one component written as:
a·x^3 + b·y^3 + c·z^3 + d·x^2 + e·y^2 + f·z^2 + g·xy + h·yz + i·xz + j·x + k·y + l·z + m
its Laplacian becomes:
6a·x + 6b·y + 6c·z + 2d + 2e + 2f
Enter the point where you want the vector Laplacian evaluated. Then fill the coefficients for each vector component P, Q, and R.
Each component follows the same polynomial template. Leave a box as zero if that term does not appear in your field.
Press Calculate Vector Laplacian. The page shows the evaluated field components, the Laplacian formulas, the final vector Laplacian, and its magnitude.
Use Load Example to test the calculator quickly. After calculation, export the summary as CSV or PDF and review the graph for x-direction behavior.
This page is built for multivariable practice, engineering math review, and classroom demonstrations. It focuses on structured polynomial vector fields, which makes the second-derivative pattern easy to inspect.
You can compare the original component formula against the Laplacian formula. That helps you see how cubic terms reduce to linear terms and how squared terms contribute constant values.
The calculator also evaluates the field itself at the chosen point. This extra step helps you compare the original vector field and the transformed vector Laplacian side by side.
The export tools are useful for homework notes, lab reports, and worked examples. The graph adds another view by sweeping x while holding y and z fixed.
The vector Laplacian applies the scalar Laplacian to each component of a vector field. For F = <P, Q, R>, it becomes <∇²P, ∇²Q, ∇²R>.
It supports a structured three-variable polynomial template with cubic, quadratic, mixed, linear, and constant terms for each component.
They vanish because the Laplacian uses second partial derivatives with respect to the same variable. The second derivative of xy, yz, or xz is zero.
The point matters when cubic terms exist. Their second derivatives produce linear terms, so the Laplacian value changes with x, y, or z.
Not exactly. A scalar Laplacian returns one value from one scalar function. A vector Laplacian returns a three-component vector from three scalar components.
This version is designed for the polynomial form shown on the page. Non-polynomial fields would need a symbolic parser or a different calculator model.
The graph sweeps the x-coordinate around your chosen point. It keeps y and z fixed and plots the three Laplacian components and magnitude.
Yes. After calculation, use the CSV button for tabular data and the PDF button for a compact printable summary.
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.