x Therefore, \[\begin{align*} curl \, \vecs{F} = (R_y - Q_z)i + (P_z - R_x)j + (Q_x - P_y)k \\[4pt] = 0. {\displaystyle \mathbf {F} =F_{x}\mathbf {i} +F_{y}\mathbf {j} +F_{z}\mathbf {k} } In this section, we examine two important operations on a vector field: divergence and curl. {\displaystyle \psi (x_{1},\ldots ,x_{n})} … Therefore, \(\vecs{F}\) satisfies the cross-partials property on a simply connected domain, and the Cross-Partial Property of Conservative Fields implies that \(\vecs{F}\) is conservative. where Just “plug and chug,” as they say. Then the divergence of V, written V.V or div V, is defined by ðx + + vak) ðz Note the analogy with A.B = Al Bl + "B2 + A3Bg. For example, the potential function of an electrostatic field in a region of space that has no static charge is harmonic. Note that the curl of a vector field is a vector field, in contrast to divergence. is always the zero vector: Here ∇2 is the vector Laplacian operating on the vector field A. n ( \[\dfrac{\partial}{\partial x} (e^x) + \dfrac{\partial}{\partial y}(yz) - \dfrac{\partial}{\partial z} (y z^2) = e^x + z - 2yz. ( Is there some formula for the Divergence of cross Product of (n-1) vectors in $\mathbb{R}_n$. Next Section . Recall that a source-free field is a vector field that has a stream function; equivalently, a source-free field is a field with a flux that is zero along any closed curve. Since the curl of the gravitational field is zero, the field has no spin. ∇ Curl 4. F Del is a formal vector; it has components, but those components have partial derivative operators (and so on) which want to be fed functions to differentiate. n Note the domain of \(\vecs{F}\) is \(\mathbb{R}^2\) which is simply connected. We have the following generalizations of the product rule in single variable calculus. Physicists use divergence in Gauss’s law for magnetism, which states that if \(\vecs{B}\) is a magnetic field, then \(\nabla \cdot \vecs{B} = 0\); in other words, the divergence of a magnetic field is zero. Divergence (Div) 3. Furthermore, \(\vecs{F}\) is continuous with differentiable component functions. f F {\displaystyle f(x)} ∇ Show Mobile Notice Show All Notes Hide All Notes. Divergence. grad x Therefore, we expect the curl of the field to be nonzero, and this is indeed the case (the curl is \(2k\)). y ) The blue circle in the middle means curl of curl exists, whereas the other two red circles (dashed) mean that DD and GG do not exist. y In part (a), the vector field is constant and there is no spin at any point. {\displaystyle \cdot } \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\), https://math.libretexts.org/@app/auth/3/login?returnto=https%3A%2F%2Fmath.libretexts.org%2FCourses%2FMount_Royal_University%2FMATH_3200%253A_Mathematical_Methods%2F9%253A_Vector_Calculus%2F9.5%253A_Divergence_and_Curl, \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\), information contact us at [email protected], status page at https://status.libretexts.org. The divergence of a vector field is a scalar function. The derivatives of scalars, vectors, and second-order tensors with respect to second-order tensors are of considerable use in continuum mechanics.These derivatives are used in the theories of nonlinear elasticity and plasticity, particularly in the design of algorithms for numerical simulations.. , also called a scalar field, the gradient is the vector field: where operations are understood not to act on the {\displaystyle \psi } Example \(\PageIndex{2}\): Determining Whether a Field Is Magnetic. Example \(\PageIndex{3}\): Determining Whether a Field Is Source Free. ⁡ ψ y In addition to defining curl and divergence, we look at some physical interpretations of them, and show their relationship to conservative and source-free vector fields. {\displaystyle \mathbf {J} _{\mathbf {B} }\,-\,\mathbf {J} _{\mathbf {B} }^{\mathrm {T} }} The divergence of the curl of any vector field (in three dimensions) is equal to zero: ∇ ⋅ ( ∇ × F ) = 0. Recall that the flux was measured via a line integral, and the sum of the divergences … of two vectors, or of a covector and a vector. Del is a formal vector; it has components, but those components have partial derivative operators (and so on) which want to be fed functions to differentiate. , Visualization of the Divergence and Curl of a vector field.My Patreon Page: https://www.patreon.com/EugeneK the curl is the vector field: where i, j, and k are the unit vectors for the x-, y-, and z-axes, respectively. , If the curl is zero, then the leaf doesn’t rotate as it moves through the fluid.