Vector Calculus Calculator

๐Ÿ“ Practice worksheet
Gradient
Enter a function of x, y, z
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Type a scalar field aboveโ€ฆ
x^2    y^3    z^2
sin(x)    cos(y)    tan(z)
e^x    e^(x*y)    exp(z)
log(x) = ln(x)    sqrt(x)
x*y    x*y*z    3*x^2
Multiplication: Use * explicitly: x*y not xy
Powers: x^2 or x^(2/3)
Constants: pi, e
โˆ‡

Enter a field and click Compute

Compute gradient, divergence, or curl with step-by-step solutions.

Compute a gradient or curl to see its 3D vector field.

What is Vector Calculus?

Vector calculus extends single-variable calculus to fields in two and three dimensions. The three fundamental operations are the gradient, divergence, and curl, all defined using the del operator โˆ‡.

These operations are essential in physics (electromagnetism, fluid dynamics, thermodynamics) and engineering (signal processing, computer graphics, robotics).

The Three Operations

OperationInputOutputFormula
Gradient โˆ‡fScalar fieldVector field(โˆ‚f/โˆ‚x, โˆ‚f/โˆ‚y, โˆ‚f/โˆ‚z)
Divergence โˆ‡ยทFVector fieldScalarโˆ‚Fx/โˆ‚x + โˆ‚Fy/โˆ‚y + โˆ‚Fz/โˆ‚z
Curl โˆ‡ร—FVector fieldVector fielddet[i,j,k; โˆ‚x,โˆ‚y,โˆ‚z; Fx,Fy,Fz]

Key Vector Calculus Identities

Curl of Gradient = 0

โˆ‡ ร— (โˆ‡f) = 0 for any smooth scalar field f. Gradient fields are always irrotational.

Divergence of Curl = 0

โˆ‡ ยท (โˆ‡ ร— F) = 0 for any smooth vector field F. Curl fields are always solenoidal.

Laplacian

โˆ‡ยฒf = โˆ‡ ยท (โˆ‡f) = โˆ‚ยฒf/โˆ‚xยฒ + โˆ‚ยฒf/โˆ‚yยฒ + โˆ‚ยฒf/โˆ‚zยฒ. The divergence of the gradient.

Divergence Theorem

&oiint; F ยท dS = &oiiint; (โˆ‡ ยท F) dV. Relates surface integral to volume integral.

Applications

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Electromagnetism

Maxwell's equations use gradient, divergence, and curl to describe electric and magnetic fields.

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Fluid Dynamics

Divergence measures fluid expansion/compression. Curl measures fluid rotation (vorticity).

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Heat Transfer

The gradient of temperature gives the direction of heat flow. Fourier's law: q = -kโˆ‡T.

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Optimization

The gradient points in the direction of steepest ascent. Gradient descent finds minima of cost functions.

Frequently asked

The gradient of a scalar field f(x,y,z) is a vector field that points in the direction of the greatest rate of increase of f. It is computed as โˆ‡f = (โˆ‚f/โˆ‚x, โˆ‚f/โˆ‚y, โˆ‚f/โˆ‚z). For example, the gradient of f = xยฒ + yยฒ + zยฒ is (2x, 2y, 2z).
The divergence of a vector field F = (Fx, Fy, Fz) is a scalar measuring the rate at which the field spreads out from a point: โˆ‡ยทF = โˆ‚Fx/โˆ‚x + โˆ‚Fy/โˆ‚y + โˆ‚Fz/โˆ‚z. Positive divergence indicates a source; negative indicates a sink.
The curl of a vector field F measures its tendency to rotate around a point. It is computed using the determinant formula involving partial derivatives. If curl F = 0 the field is conservative (irrotational). For example, curl(โˆ’y, x, 0) = (0, 0, 2), indicating uniform rotation.
Use standard math notation: x^2 for squared, sin(x), e^z, sqrt(x), x*y for multiplication. Polynomials, trig, exponential, log, and hyperbolic functions all work. A live KaTeX preview shows your expression as rendered math.
Yes โ€” after computing a result, click Show Steps for a detailed solution showing each partial derivative, simplification, and final assembly of the result vector. Steps are rendered with full LaTeX math notation.
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