Vector Calculus
Essential formulas and concepts in vector calculus
Gradient (of a scalar field)
\(\displaystyle \nabla f = \Big(\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z}\Big)\)
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Divergence (of a vector field)
\(\displaystyle \nabla\cdot \mathbf{F} = \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} + \frac{\partial R}{\partial z}\)
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Divergence is a scalar field representing the net rate of "outflow" from an infinitesimal volume at each point. Positive divergence indicates a source (flow expanding outward), negative indicates a sink.
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Curl (of a vector field)
\(\displaystyle \nabla \times \mathbf{F} = \Big(\frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z}, \frac{\partial P}{\partial z} - \frac{\partial R}{\partial x}, \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\Big)\)
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Laplacian
\(\displaystyle \nabla^2 f = \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} + \frac{\partial^2 f}{\partial z^2}\)
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- \(f is a twice-differentiable scalar function of x,y,z.\)
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Green’s Theorem (in the plane)
\(\displaystyle \iint_{D}\Big(\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\Big)\,dA = \oint_{\partial D} \big(P\,dx + Q\,dy\big)\)
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Relates a double integral over region \(D\) to a line integral around its boundary. It’s a special case of Stokes’ Theorem in 2D, often used to convert between circulation (line integral) and curl (area integral).
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Stokes’ Theorem
\(\displaystyle \iint_{S} (\nabla \times \mathbf{F})\cdot d\mathbf{S} = \oint_{\partial S} \mathbf{F}\cdot d\mathbf{\ell}\)
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Divergence Theorem (Gauss’s Theorem)
\(\displaystyle \iiint_{V} (\nabla \cdot \mathbf{F})\,dV = \iint_{\partial V} \mathbf{F}\cdot d\mathbf{A}\)
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Relates the flux of a vector field out of a closed surface to the volume integral of the divergence over the region inside. In essence, it formalizes that the total "outflow" from a volume equals the sum of all sources inside.
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