Vector Calculus Solver

Partial Derivative & Double Integral Calculator

Vector calculus extends the calculus you learned in single-variable form into two and three dimensions. This solver handles partial derivatives (∂f/∂x, ∂f/∂y, ∂f/∂z), gradients ∇f, double integrals over rectangular and polar regions, divergence ∇·F, and curl ∇×F — with full step-by-step working.

How to use the partial derivative calculator

• Type \frac{\partial}{\partial x}(x^2 y + 3xy^2) for ∂/∂x
• Type \frac{\partial}{\partial y}(x^2 y + 3xy^2) for ∂/∂y
• For gradient: \nabla(x^2 + y^2 + z^2) returns ⟨2x, 2y, 2z⟩

What is a partial derivative?

For a multi-variable function f(x, y, z, ...), a partial derivative is the rate of change with respect to one variable, while all the others are held constant. So if f(x, y) = x²y + 3xy²:

Notation: ∂f/∂x is read "partial f partial x". The curly ∂ distinguishes it from the ordinary d/dx of single-variable calculus. Used heavily in physics (gradient, divergence), economics (marginal analysis), and machine learning (backpropagation).

How to use the double integral calculator

• Type \int_0^1 \int_0^2 xy \, dx \, dy for ∬xy over [0,2]×[0,1]
• Type \int_0^\pi \int_0^1 r \, dr \, d\theta for the area of a half-disc in polar form

Double integral — worked example

Evaluate ∬_R xy dA where R is the rectangle [0, 2] × [0, 1].

• Step 1. Set up: ∫₀¹ ∫₀² xy dx dy.
• Step 2. Inner integral (treat y as constant): ∫₀² xy dx = y · [x²/2]₀² = y · 2 = 2y.
• Step 3. Outer integral: ∫₀¹ 2y dy = [y²]₀¹ = 1.
• Final answer: 1.

Polar double integrals

When the region of integration is a disc or annulus, switching to polar coordinates (r, θ) often simplifies the problem. The area element changes: dA = r dr dθ (the extra r is the Jacobian). So the area of a disc of radius R:

Gradient, divergence, curl — quick reference

Frequently Asked Questions