Cylindrical coordinates

The cylindrical coordinate system is very convenient when dealing with problems that have cylindrical or azimuthal symmetry. It is a three-dimensional version of the polar coordinates of plane analytic geometry.

A point P{displaystyle P} in cylindrical coordinates is represented by (ρ ρ ,φ φ ,z){displaystyle (rhovarphiz)} where:

• ρ ρ {displaystyle rho }: Coordinated radial, defined as the point distance P{displaystyle P} axis z{displaystyle z}or the length of the radio projection on the plane XAnd{displaystyle XY}
• φ φ {displaystyle varphi }: Coordinated azimutaldefined as the angle that forms with the shaft X{displaystyle X} the radiovector projection on the plane XAnd{displaystyle XY}.
• z{displaystyle z}: Coordinated vertical or height, defined as the distance, with sign, from point P to plane XAnd{displaystyle XY}.

The ranges of variation of the three coordinates are

<math alttext="{displaystyle 0leq rho <infty qquad 0leq varphi <2pi qquad -infty <z0≤ ≤ ρ ρ .∞ ∞ 0≤ ≤ φ φ .2π π − − ∞ ∞ .z.∞ ∞ {displaystyle 0leq rho liceinfty qquad 0leq varphi ≤2pi qquad -infty ≤ ≤ ¢Üinfty }<img alt="0leq rho <infty qquad 0leq varphi <2pi qquad -infty <z

The azimutal coordinate φ φ {displaystyle varphi } on occasion − − π π {displaystyle} a π π {displaystyle pi }. The radial coordinate is always positive. If you reduce the value of ρ ρ {displaystyle rho } the value 0 is reached from there, ρ ρ {displaystyle rho } increase again, but φ φ {displaystyle varphi } increase or decrease in π π {displaystyle pi } radian.

Relation to other coordinate systems

Relation to Cartesian coordinates

Cylindrical coordinates and related Cartesian axis.

Taking into account the definition of the angle φ φ {displaystyle varphi }, we get the following relationships between the cylindrical coordinates and the cartesianas:

x=ρ ρ # φ φ ,and=ρ ρ without φ φ ,z=z{displaystyle x=rho cos varphiqquad y=rho sin varphiqquad z=z}

Lines and coordinate surfaces

Coordinate lines are those obtained by varying one of the coordinates and keeping the other two fixed. For cylindrical coordinates, these are:

• Lines coordinates ρ: Horizontal semi-recipes starting from the axis Z{displaystyle Z}.
• Coordinated lines φ φ {displaystyle varphi }: Horizontal circles.
• Coordinated lines z{displaystyle z}: Vertical recipes.

The coordinate surfaces are those that are obtained by successively fixing each of the coordinates of a point. For this system they are:

• Surfaces ρ=cte.: Vertical straight cylinders.
• Areas φ φ {displaystyle varphi }=cte.: Vertical Semiplanes.
• Areas z{displaystyle z}=cte.: Horizontal plans.

The coordinate lines and surfaces of this system are perpendicular two by two at each point. Therefore, this is an orthogonal system.

Coordinate base

From the cylindrical coordinate system, a vector base can be defined at each point in space, by means of the tangent vectors to the coordinate lines. This new basis can be related to the fundamental basis of Cartesian coordinates by the relations

ρ ρ ^ ^ =# φ φ x^ ^ +senφ φ and^ ^ {displaystyle {hat {rho }}=cos varphi ,{hat {x}+}{rm {sen}}},varphi ,{hat {y}}}}}}}

φ φ ^ ^ =− − senφ φ x^ ^ +#φ φ and^ ^ {displaystyle {hat {varphi}}=-{rm {sen}}}varphi ,{hat {x}+cos ,varphi ,{hat {y}}}}}}}

z^ ^ =z^ ^ {displaystyle {hat {z}}={hat {z}}}

and vice versa

x^ ^ =# φ φ ρ ρ ^ ^ − − senφ φ φ φ ^ ^ {displaystyle {hat {x}}=cos varphi ,{hat {rho }-}{rm {sen}}}{varphi ,{hat {varphi }}}}}}}}

and^ ^ =senφ φ ρ ρ ^ ^ +#φ φ φ φ ^ ^ {displaystyle {hat {y}}={rm {sen}}varphi ,{hat {rho}}+cos ,varphi ,{hat {varphi }}}}}}}

z^ ^ =z^ ^ {displaystyle {hat {z}}={hat {z}}}

In the calculation of this base, the scale factors are obtained

hρ ρ =1hφ φ =ρ ρ hz=1{displaystyle h_{rho }=1qquad h_{varphi }=rho qquad h_{z}=1}

Having the cylindrical coordinate base, it is obtained that the expression of the position vector in these coordinates is

r→ → =ρ ρ ρ ρ ^ ^ +zz^ ^ {displaystyle {vec {r}}=rho ,{hat {rho }+z,{hat {z}}}}

Note that a term does not appear φ φ φ φ ^ ^ {displaystyle varphi ,{hat {varphi }}}. The dependence on this coordinate is hidden hidden hidden hidden hidden hidden hidden hidden hidden hidden hidden hidden hidden hidden hidden hidden hidden hidden hidden hidden hidden hidden hidden hidden hidden hidden hidden hidden hidden hidden hidden hidden hidden hidden hidden hidden hidden in the vectors of the base.

Indeed:

r→ → =xı ı ^ ^ +and ^ ^ +zk^ ^ =ρ ρ # φ φ ı ı ^ ^ +ρ ρ without φ φ ^ ^ +zk^ ^ =ρ ρ (# φ φ ı ı ^ ^ +without φ φ ^ ^ )+zk^ ^ =ρ ρ ρ ρ ^ ^ +zz^ ^ {cHFFFF}{cHFFFF}{cHFFFFFF}{cHFFFF}{cHFFFF}{cHFFFFFF}{cHFFFFFF}{cHFFFFFF}{cHFFFF}{cHFFFF}{cHFFFF}{cHFFFF}{cHFF}{cHFF}{cH00}{cHFFFF}{cH00}{cH00}{cHFFFFFFFFFFFF}{cH00}{cHFFFFFFFFFFFFFF}{cH00} {cHFFFFFFFFFFFFFFFF}{cH00}{cH00}{cH00}{cH00}{cHFFFFFF}{cHFFFFFF}{cHFF} {cHFFFFFFFFFFFFFFFFFF}{cH00}{cH00}{cH00} {cHFF

Line, area and volume spreads

Line differential

An infinitesimal displacement, expressed in cylindrical coordinates, is given by

dr→ → =hρ ρ dρ ρ ρ ρ ^ ^ +hφ φ dφ φ φ φ ^ ^ +hzdzz^ ^ =dρ ρ ρ ρ ^ ^ +ρ ρ dφ φ φ φ ^ ^ +dzz^ ^ {displaystyle d{vec {r}}=h_{rho },drho ,{hat {rho }} +h_{varphi },dvarphi }{,{hat {varphi}}{varf}}{,{cHFF}{cHFF}{cHFFFF}{cHFFFFFFFFFF}{cHFF}{cHFF}{cHFFFFFFFFFF}{cHFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF}{cHFF}{cHFF}{cHFFFFFFFF}{cHFF}{cHFFFFFF}{cHFF}{cHFF}{cHFFFF}{cHFF}{cHFFFF}{cHFF}{cHFFFFFF}{cHFFFFFFFF}{cHFFFFFF}{cHFF}{

Surface differentials

The general expression of a surface differential in curvilinear coordinates is complicated.

However, for the case of a coordinated surface, q3=cte.{displaystyle q_{3}={rm {cte}}}} the result is

dS→ → q3=cte=h1h2dq1dq2q^ ^ 3{displaystyle d{vec {S}_{q_{3}={rm {cte}}}}=h_{1},h_{2},dq_{1},dq_{2},{hat {q}}}{3}}}}

and analogous expressions for the other two coordinate surfaces.

In the particular case of cylindrical coordinates, the surface differentials are

• ρ=cte: dS→ → ρ ρ =cte=ρ ρ dφ φ dzρ ρ ^ ^ {displaystyle d{vec {S}_{rho ={rm {cte}=}{rho ,dvarphi ,dz,{hat {rho }}}}}}}
• φ=cte: dS→ → φ φ =cte=dρ ρ dzφ φ ^ ^ {displaystyle d{vec {S}_{varphi ={rm {cte}}=drho ,dz,{hat {varphi }}}}}}
• z=cte: dS→ → z=cte=ρ ρ dρ ρ dφ φ z^ ^ {displaystyle d{vec {S}_{z={rm {cte}}}=rho ,dvarphi ,{hat {z}}}}}}

Volume spread

The volume of an element in curvilinear coordinates is equal to the product of the Jacobian of the transformation, multiplied by the three differentials. The Jacobian, in turn, is equal to the product of the three scale factors, so

dV=h1h2h3dq1dq2dq3{displaystyle dV=h_{1},h_{2},h_{3},dq_{1},dq_{2},dq_{3}}}}}}

which for cylindrical coordinates gives

dV=ρ ρ dρ ρ dφ φ dz{displaystyle dV=rho ,drho ,dvarphi ,dz}

Differential operators in cylindrical coordinates

The gradient, the divergence, the curl and the Laplacian have particular expressions in cylindrical coordinates. These are:

• Gradient

► ► φ φ =▪ ▪ φ φ ▪ ▪ ρ ρ ρ ρ ^ ^ +1ρ ρ ▪ ▪ φ φ ▪ ▪ φ φ φ φ ^ ^ +▪ ▪ φ φ ▪ ▪ zz^ ^ {displaystyle nabla phi ={frac {partial phi }{partial rho }}{hat {rho }+{frac {1}{rho }}{frac}{partial phi }{partial varphi }}{partial varphi }{varhat {varphi }{

• Divergence

► ► ⋅ ⋅ F→ → =1ρ ρ ▪ ▪ (ρ ρ Fρ ρ )▪ ▪ ρ ρ +1ρ ρ ▪ ▪ Fφ φ ▪ ▪ φ φ +▪ ▪ Fz▪ ▪ z{displaystyle nabla cdot {vec {F}}={frac {1{rho }}{frac {partial}{rho F_{rho }}}{partial rho }}{frac {1}{rho }{fpartial F_partial}{varphi }}{

• Rotation

► ► × × F→ → =1ρ ρ 日本語ρ ρ ^ ^ ρ ρ φ φ ^ ^ z^ ^ ▪ ▪ ▪ ▪ ρ ρ ▪ ▪ ▪ ▪ φ φ ▪ ▪ ▪ ▪ zFρ ρ ρ ρ Fφ φ Fz日本語=ρ ρ ^ ^ (1ρ ρ ▪ ▪ Fz▪ ▪ φ φ − − ▪ ▪ Fφ φ ▪ ▪ z)+φ φ ^ ^ (▪ ▪ Fρ ρ ▪ ▪ z− − ▪ ▪ Fz▪ ▪ ρ ρ )+z^ ^ [chuckles]1ρ ρ ▪ ▪ (ρ ρ Fφ φ )▪ ▪ ρ ρ − − 1ρ ρ ▪ ▪ Fρ ρ ▪ ▪ φ φ ]## #################################################################################

• Laplaciano

► ► 2φ φ =1ρ ρ ▪ ▪ ▪ ▪ ρ ρ (ρ ρ ▪ ▪ φ φ ▪ ▪ ρ ρ )+1ρ ρ 2▪ ▪ 2φ φ ▪ ▪ φ φ 2+▪ ▪ 2φ φ ▪ ▪ z2{displaystyle nabla ^{2}phi ={frac {1}{rho }{frac}{partial }{partial rho }{left(rho}{frac}{partial phi }{partial rho }{rho }}{