Gradient, divergence, curl, and other vector calculus terms and what each operator measures. Front: the term. Back: a plain-language definition.
36 cards · basic cards · AI-written, checked twice. Edit anything.
- Gradient
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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
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A scalar measure of how much a vector field spreads out or converges at a point, representing the net outflow per unit volume.- Curl
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A vector operator that measures the rotation or circulation tendency of a vector field at a point.- Laplacian
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The divergence of the gradient of a function, written as the sum of second partial derivatives with respect to each coordinate.- Directional derivative
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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
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The symbol (upside-down triangle) used to denote gradient, divergence, and curl, formally a vector of partial derivatives.- Vector field
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A function that assigns a vector to each point in space, such as a velocity field or a magnetic field.- Scalar field
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A function that assigns a scalar (single number) to each point in space, such as temperature or pressure.- Conservative vector field
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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
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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
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A vector field whose curl is zero everywhere, meaning it has no rotation or circulation.- Solenoidal field
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A vector field whose divergence is zero everywhere, meaning there are no sources or sinks of the field.- Line integral
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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
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An integral of a function over a surface in space, used to compute quantities like flux of a field through the surface.- Circulation
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The line integral of a vector field around a closed curve, measuring the total rotation or swirl of the field.
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