Gradient, divergence, curl, and other vector calculus terms and what each operator measures. Front: the term. Back: a plain-language definition.
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- Gradient
- A vector operator that points in the direction of maximum increase of a scalar function, with magnitude equal to the rate of increase in that direction.
- Divergence
- A scalar measure of how much a vector field spreads out or converges at a point, representing the net outflow per unit volume.
- Curl
- A vector operator that measures the rotation or circulation tendency of a vector field at a point.
- Laplacian
- The divergence of the gradient of a function, written as the sum of second partial derivatives with respect to each coordinate.
- Directional derivative
- The rate of change of a scalar function in a specified direction, equal to the dot product of the gradient with a unit direction vector.
- Nabla or del operator
- The symbol (upside-down triangle) used to denote gradient, divergence, and curl, formally a vector of partial derivatives.
- Vector field
- A function that assigns a vector to each point in space, such as a velocity field or a magnetic field.
- Scalar field
- A function that assigns a scalar (single number) to each point in space, such as temperature or pressure.
- Conservative vector field
- A vector field that can be written as the gradient of a scalar potential function, with the property that line integrals are path-independent.
- Potential function
- A scalar function f such that a given vector field F equals the negative gradient of f, often denoted F = negative grad f.
- Irrotational field
- A vector field whose curl is zero everywhere, meaning it has no rotation or circulation.
- Solenoidal field
- A vector field whose divergence is zero everywhere, meaning there are no sources or sinks of the field.
- Line integral
- An integral of a function along a curve in space, used to compute quantities like work done by a force along a path.
- Surface integral
- An integral of a function over a surface in space, used to compute quantities like flux of a field through the surface.
- Circulation
- The line integral of a vector field around a closed curve, measuring the total rotation or swirl of the field.