This is a list of some vector calculus formulae for working with common curvilinear coordinate systems.
Notes This article uses the standard notation ISO 80000-2, which supersedes ISO 31-11, for spherical coordinates (other sources may reverse the definitions of θ and φ): The polar angle is denoted by θ ∈ [ 0 , π ] {\displaystyle \theta \in [0,\pi ]} : it is the angle between the z-axis and the radial vector connecting the origin to the point in question. The azimuthal angle is denoted by φ ∈ [ 0 , 2 π ] {\displaystyle \varphi \in [0,2\pi ]} : it is the angle between the x-axis and the projection of the radial vector onto the xy-plane. The function atan2(y, x) can be used instead of the mathematical function arctan(y/x) owing to its domain and image. The classical arctan function has an image of (−π/2, +π/2), whereas atan2 is defined to have an image of (−π, π].
Coordinate conversions
Note that the operation arctan ( A B ) {\displaystyle \arctan \left({\frac {A}{B}}\right)} must be interpreted as the two-argument inverse tangent, atan2.
Unit vector conversions
Del formula
Differential elements
Calculation rules
div grad f ≡ ∇ ⋅ ∇ f ≡ ∇ 2 f {\displaystyle \operatorname {div} \,\operatorname {grad} f\equiv \nabla \cdot \nabla f\equiv \nabla ^{2}f}
curl grad f ≡ ∇ × ∇ f = 0 {\displaystyle \operatorname {curl} \,\operatorname {grad} f\equiv \nabla \times \nabla f=\mathbf {0} }
div curl A ≡ ∇ ⋅ ( ∇ × A ) = 0 {\displaystyle \operatorname {div} \,\operatorname {curl} \mathbf {A} \equiv \nabla \cdot (\nabla \times \mathbf {A} )=0}
curl curl A ≡ ∇ × ( ∇ × A ) = ∇ ( ∇ ⋅ A ) − ∇ 2 A {\displaystyle \operatorname {curl} \,\operatorname {curl} \mathbf {A} \equiv \nabla \times (\nabla \times \mathbf {A} )=\nabla (\nabla \cdot \mathbf {A} )-\nabla ^{2}\mathbf {A} } (Lagrange's formula for del)
∇ 2 ( f g ) = f ∇ 2 g + 2 ∇ f ⋅ ∇ g + g ∇ 2 f {\displaystyle \nabla ^{2}(fg)=f\nabla ^{2}g+2\nabla f\cdot \nabla g+g\nabla ^{2}f}
∇ 2 ( P ⋅ Q ) = Q ⋅ ∇ 2 P − P ⋅ ∇ 2 Q + 2 ∇ ⋅ [ ( P ⋅ ∇ ) Q + P × ∇ × Q ] {\displaystyle \nabla ^{2}\left(\mathbf {P} \cdot \mathbf {Q} \right)=\mathbf {Q} \cdot \nabla ^{2}\mathbf {P} -\mathbf {P} \cdot \nabla ^{2}\mathbf {Q} +2\nabla \cdot \left[\left(\mathbf {P} \cdot \nabla \right)\mathbf {Q} +\mathbf {P} \times \nabla \times \mathbf {Q} \right]\quad } (From )
Cartesian derivation
div A = lim V → 0 ∬ ∂ V A ⋅ d S ∭ V d V = [ A x ( x + d x ) − A x ( x ) ] d y d z + [ A y ( y + d y ) − A y ( y ) ] d x d z + [ A z ( z + d z ) − A z ( z ) ] d x d y d x d y d z = ∂ A x ∂ x + ∂ A y ∂ y + ∂ A z ∂ z {\displaystyle {\begin{aligned}\operatorname {div} \mathbf {A} &=\lim _{V\to 0}{\frac {\iint _{\partial V}\mathbf {A} \cdot d\mathbf {S} }{\iiint _{V}dV}}\\[1ex]&={\frac {\left[A_{x}(x{+}dx)-A_{x}(x)\right]dy\,dz+\left[A_{y}(y{+}dy)-A_{y}(y)\right]dx\,dz+\left[A_{z}(z{+}dz)-A_{z}(z)\right]dx\,dy}{dx\,dy\,dz}}\\&={\frac {\partial A_{x}}{\partial x}}+{\frac {\partial A_{y}}{\partial y}}+{\frac {\partial A_{z}}{\partial z}}\end{aligned}}}
( curl A ) x = lim S ⊥ x ^ → 0 ∫ ∂ S A ⋅ d ℓ ∬ S d S = [ A z ( y + d y ) − A z ( y ) ] d z − [ A y ( z + d z ) − A y ( z ) ] d y d y d z = ∂ A z ∂ y − ∂ A y ∂ z {\displaystyle {\begin{aligned}(\operatorname {curl} \mathbf {A} )_{x}&=\lim _{S^{\perp \mathbf {\hat {x}} }\to 0}{\frac {\int _{\partial S}\mathbf {A} \cdot d\mathbf {\ell } }{\iint _{S}dS}}\\[1ex]&={\frac {\left[A_{z}(y{+}dy)-A_{z}(y)\right]dz-\left[A_{y}(z{+}dz)-A_{y}(z)\right]dy}{dy\,dz}}\\&={\frac {\partial A_{z}}{\partial y}}-{\frac {\partial A_{y}}{\partial z}}\end{aligned}}}
The expressions for ( curl A ) y {\displaystyle (\operatorname {curl} \mathbf {A} )_{y}} and ( curl A ) z {\displaystyle (\operatorname {curl} \mathbf {A} )_{z}} are found in the same way.
Cylindrical derivation
div A = lim V → 0 ∬ ∂ V A ⋅ d S ∭ V d V = [ A ρ ( ρ + d ρ ) ( ρ + d ρ ) − A ρ ( ρ ) ρ ] d ϕ d z + [ A ϕ ( ϕ + d ϕ ) − A ϕ ( ϕ ) ] d ρ d z + [ A z ( z + d z ) − A z ( z ) ] d ρ ( ρ + d ρ / 2 ) d ϕ ρ d ϕ d ρ d z = 1 ρ ∂ ( ρ A ρ ) ∂ ρ + 1 ρ ∂ A ϕ ∂ ϕ + ∂ A z ∂ z {\displaystyle {\begin{aligned}\operatorname {div} \mathbf {A} &=\lim _{V\to 0}{\frac {\iint _{\partial V}\mathbf {A} \cdot d\mathbf {S} }{\iiint _{V}dV}}\\&={\frac {\left[A_{\rho }(\rho {+}d\rho )(\rho {+}d\rho )-A_{\rho }(\rho )\rho \right]d\phi \,dz+\left[A_{\phi }(\phi {+}d\phi )-A_{\phi }(\phi )\right]d\rho \,dz+\left[A_{z}(z{+}dz)-A_{z}(z)\right]d\rho (\rho {+}d\rho /2)\,d\phi }{\rho \,d\phi \,d\rho \,dz}}\\&={\frac {1}{\rho }}{\frac {\partial (\rho A_{\rho })}{\partial \rho }}+{\frac {1}{\rho }}{\frac {\partial A_{\phi }}{\partial \phi }}+{\frac {\partial A_{z}}{\partial z}}\end{aligned}}}
( curl A ) ρ = lim S ⊥ ρ ^ → 0 ∫ ∂ S A ⋅ d ℓ ∬ S d S = A ϕ ( z ) ( ρ + d ρ ) d ϕ − A ϕ ( z + d z ) ( ρ + d ρ ) d ϕ + A z ( ϕ + d ϕ ) d z − A z ( ϕ ) d z ( ρ + d ρ ) d ϕ d z = − ∂ A ϕ ∂ z + 1 ρ ∂ A z ∂ ϕ {\displaystyle {\begin{aligned}(\operatorname {curl} \mathbf {A} )_{\rho }&=\lim _{S^{\perp {\hat {\boldsymbol {\rho }}}}\to 0}{\frac {\int _{\partial S}\mathbf {A} \cdot d{\boldsymbol {\ell }}}{\iint _{S}dS}}\\[1ex]&={\frac {A_{\phi }(z)\left(\rho +d\rho \right)\,d\phi -A_{\phi }(z+dz)\left(\rho +d\rho \right)\,d\phi +A_{z}(\phi +d\phi )\,dz-A_{z}(\phi )\,dz}{\left(\rho +d\rho \right)\,d\phi \,dz}}\\[1ex]&=-{\frac {\partial A_{\phi }}{\partial z}}+{\frac {1}{\rho }}{\frac {\partial A_{z}}{\partial \phi }}\end{aligned}}}
( curl A ) ϕ = lim S ⊥ ϕ ^ → 0 ∫ ∂ S A ⋅ d ℓ ∬ S d S = A z ( ρ ) d z − A z ( ρ + d ρ ) d z + A ρ ( z + d z ) d ρ − A ρ ( z ) d ρ d ρ d z = − ∂ A z ∂ ρ + ∂ A ρ ∂ z {\displaystyle {\begin{aligned}(\operatorname {curl} \mathbf {A} )_{\phi }&=\lim _{S^{\perp {\boldsymbol {\hat {\phi }}}}\to 0}{\frac {\int _{\partial S}\mathbf {A} \cdot d{\boldsymbol {\ell }}}{\iint _{S}dS}}\\&={\frac {A_{z}(\rho )\,dz-A_{z}(\rho +d\rho )\,dz+A_{\rho }(z+dz)\,d\rho -A_{\rho }(z)\,d\rho }{d\rho \,dz}}\\&=-{\frac {\partial A_{z}}{\partial \rho }}+{\frac {\partial A_{\rho }}{\partial z}}\end{aligned}}}
( curl A ) z = lim S ⊥ z ^ → 0 ∫ ∂ S A ⋅ d ℓ ∬ S d S = A ρ ( ϕ ) d ρ − A ρ ( ϕ + d ϕ ) d ρ + A ϕ ( ρ + d ρ ) ( ρ + d ρ ) d ϕ − A ϕ ( ρ ) ρ d ϕ ρ d ρ d ϕ = − 1 ρ ∂ A ρ ∂ ϕ + 1 ρ ∂ ( ρ A ϕ ) ∂ ρ {\displaystyle {\begin{aligned}(\operatorname {curl} \mathbf {A} )_{z}&=\lim _{S^{\perp {\hat {\boldsymbol {z}}}}\to 0}{\frac {\int _{\partial S}\mathbf {A} \cdot d\mathbf {\ell } }{\iint _{S}dS}}\\[1ex]&={\frac {A_{\rho }(\phi )\,d\rho -A_{\rho }(\phi +d\phi )\,d\rho +A_{\phi }(\rho +d\rho )(\rho +d\rho )\,d\phi -A_{\phi }(\rho )\rho \,d\phi }{\rho \,d\rho \,d\phi }}\\[1ex]&=-{\frac {1}{\rho }}{\frac {\partial A_{\rho }}{\partial \phi }}+{\frac {1}{\rho }}{\frac {\partial (\rho A_{\phi })}{\partial \rho }}\end{aligned}}}
curl