Del in cylindrical and spherical coordinates

Encyclopedia : D : DE : DEL : Del in cylindrical and spherical coordinates

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This is a list of some vector calculus formulae of general use in working with standard coordinate systems.

Table with the del operator in cylindrical and spherical coordinates

Operation	Cartesian coordinates (x,y,z)	Cylindrical coordinates (ρ,φ,z)	Spherical coordinates (r,θ,φ)
Definition of coordinates		[\left[\begin x & = & \rho\cos\phi \\ y & = & \rho\sin\phi \\ z & = & z \end\right.]	[\left[\begin x & = & r\sin\theta\cos\phi \\ y & = & r\sin\theta\sin\phi \\ z & = & r\cos\theta \end\right.]
		[\left[\begin \rho & = & \sqrt \\ \phi & = & \operatorname(y, x) \\ z & = & z \end\right.]	[\left[begin r & = & sqrt \ theta & = & arccos(z / r) \ phi & = & operatorname(y, x) endright.]
[mathbf]	[A_xmathbf + A_ymathbf + A_zmathbf]	[A_rhoboldsymbol + A_phiboldsymbol + A_zboldsymbol]	[A_rboldsymbol + A_thetaboldsymbol + A_phiboldsymbol]
[nabla f]	[mathbf + mathbf + mathbf]	[boldsymbol + boldsymbol + boldsymbol]	[boldsymbol + boldsymbol + boldsymbol]
[nabla cdot mathbf]	[ + + ]	[ + + ]	[ + ( A_thetasintheta ) + ]
[nabla times mathbf]	[begin ( - ) mathbf & + \ ( - ) mathbf & + \ ( - ) mathbf & end]	[begin ( - ) boldsymbol & + \ ( - ) boldsymbol & + \ ( - ) boldsymbol & end]	[begin ( ( A_phisintheta ) - ) boldsymbol & + \ ( - ( r A_phi ) ) boldsymbol & + \ ( ( r A_theta ) - ) boldsymbol & end]
[Delta f = nabla^2 f]	[ + + ]	[(rho ) + + ]	[(r^2 ) + (sintheta ) + ]
[Delta mathbf = nabla^2 mathbf]	[Delta A_x mathbf + Delta A_y mathbf + Delta A_z mathbf ]	[begin (Delta A_rho - - ) boldsymbol & + \ (Delta A_phi - + ) boldsymbol & + \ (Delta A_z ) boldsymbol & end]	[begin (Delta A_r - - - - ) boldsymbol & + \ (Delta A_theta - + - ) boldsymbol & + \ (Delta A_phi - + + ) boldsymbol & end]
Differential displacement	[dmathbf = dxmathbf + dymathbf + dzmathbf]	[dmathbf = drhoboldsymbol + rho dphiboldsymbol + dzboldsymbol]	[dmathbf = drmathbf + rdthetaboldsymbol + rsintheta dphiboldsymbol]
Differential normal area	[begindmathbf = &dydzmathbf + \ &dxdzmathbf + \ &dxdymathbfend]	[begindmathbf = & rho dphi dzboldsymbol + \ & drho dzboldsymbol + \ & rho drho dphi mathbf end]	[begindmathbf = & r^2 sintheta dtheta dphi mathbf + \& rsintheta drdphi boldsymbol + \& rdrdthetaboldsymbol end]
Differential volume	[dv = dxdydz ,]	[dv = rho drho dphi dz,]	[dv = r^2sintheta drdtheta dphi,]
Non-trivial calculation rules: [operatorname f = nabla cdot (nabla f) = nabla^2 f = Delta f] (Laplacian) [operatorname f = nabla times (nabla f) = 0] [operatorname mathbf = nabla cdot (nabla times mathbf) = 0] [operatorname mathbf = nabla times (nabla times mathbf) = nabla (nabla cdot mathbf) - nabla^2 mathbf] [Delta f g = f Delta g + 2 nabla f cdot nabla g + g Delta f] Lagrange's formula for the cross product: [mathbf times (mathbf times mathbf) = mathbf (mathbf cdot mathbf) - mathbf (mathbf cdot mathbf)]

• Note: This page uses standard physics notation; some (American mathematics) sources define θ as the angle with the [xy ]-plane instead of [phi].
• Note: The function atan2(y, x) is used instead of the mathematical function arctan(y/x) due to its domain and image. The classical arctan(y/x) has an image of (-π/2, +π/2), whereas atan2(y, x) is defined to have an image of (-π, π].

See also

• Curvilinear coordinates
• Vector fields in cylindrical and spherical coordinates

 

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