Flashcards in 7. Curvilinear Coordinate Systems Deck (25):

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##
Curvilinear

Definition

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-in performing volume and surface integrals we have often used non-Cartesian coosdinate systems in which the coordinate lines and surfaces are curved

-such coordinate systems are called curvilinear

e.g. cylindrical and spherical polars

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##
Curviliear Unit Vector

Definition

###
-consider a general curvilinear coordinate system (u,v,w) given locally by:

x = x(u,v,w) , y = y(u,v,w) and z = z(u,v,w)

-consider an infinitesimal displacement:

d|x = (dx, dy, dz)

-using the chain rule for partial derivatives:

d|x = (∂|x/∂u du , ∂|x/∂v dv , ∂|x/∂w dw)

-sub in tangent vectors: tu, tv, tw :

d|x = (tu du , tv dv , tw dw)

-we can define a unit vector in the u direction by

|eu = tu / |tu| , where tu is the tangent vector tu=∂|x/∂u

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##
Curvilinear Scale Factor

Definition

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-given a unit vector in the u direction in terms of a tangent vector tu :

|eu = tu / |tu|

-the associated scale factor hu is the magnitude of tu:

hu = |tu|

-so the unit vector is

|eu = tu / hu

-and a small change in displacement is given by:

d|x = hu eu du + hv ev dv + hw ew dw

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## When can a set of unit vectors be used to express a vector field?

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-provided eu . (ev x ew) ≠ 0, we can express vector fields in terms of the basis vectors eu, ev and ew

-this formula gives the volume of a parallelepiped formed by the three tangent vectors tu, tv and tw

-so long as none of the vectors are coplanar, the volume will not be zero

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## Orthogonal Curviliear Coordinates

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-although in principal we can use any system of coordinates for which eu . (ev x ew) ≠ 0

-in practice it is much easier to use systems in which eu, ev and ew are mutually perpendicular

i.e. where:

eu . ev = eu . ew = ev . ew = 0

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##
Curviliear Coordinates

Left vs Right Handed Basis

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-if we choose an orthogonal coordinate system, the basis will be either left or right handed

-if, eu . (ev x ew) = 1 then the basis is right handed and obeys the right hand rule

-if eu . (ev x ew) = -1 then the basis is left handed and obeys the left hand rule

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## Vector Algebra for Right-Handed Orthogonal Coordinate Systems

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-since ^eu, ^ev and ^ew satisfy the same relationships with scalar and vector products as ^e1, ^e2 and ^e3, i.e.

^ei ^ej = 𝛿ij

^ej x ^ek = εijk ^ei

-the rules of vector algebra are identical in the (u,v,w) coordinate system

-so dot products are calculated by summing the products of the ith components of two vectors and cross products can be calculated by finding a determinant if the e1 e2 and e3 unit vectors are replaced with the new basis vectors

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## Vector Differentiation for Right-Handed Orthogonal Coordinate Systems

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-vector differentiation does not treat the components in the same way as Cartesian components, since ^eu, ^ev and ^ew vary with position

-thus we need to find general formulae for grad, div curl and ∇²

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## The Jacobian for Volume Integrals for Right-Handed Coordinates

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|J| = h1 h2 h3

dV = h1 h2 h3 dudvdw

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## The Jacobian for Surface Integrals for Right-Handed Coordinates

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-consideran element of surface with normal in the positive ^eu direction

-this is a surface of constant u so we can paramaterise it using v and w so that:

d|S = (|tv x |tw) dv dw = hv hw (^ev x ^ew) dvdw

-but, ^eu = ^ev x ^ew , so:

d|S = hv hw ^eu dvdw

and

dS = hv hw dv dw

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##
Finding Basis Vectors

Method

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1) write the position vector |x = (x,y,z) in terms of the new coordinate system

2 )find tangent vectors for each new variable

e.g. |tr = d|x / dr

3) find the correspondig scale factor h

e.g. hr = | |tr |

4) the basis vector is the unit vector in the direction of the tangent vector

e.g. ^er = |tr / hr

5) check orthogonality by finding dot products between each of the new basis vectors (these should all be zero)

6) check handedness with scalar triple product, this should be either 1 (right handed) or -1 (left handed)

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## General Equation for an Ellipse

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x²/A² + y²/B² = 1

with A and B constant

13

## Can we use a stretched version of cylindrical polars to model constant elliptical surface?

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-a stretched version of polar coordinates:

x = aRcosφ , y = bRsinφ , z = z , so that surfaces of constant R are elliptical

-however this transformation also means that surfaces of constant R are no longer perpendicular to surfaces of constant φ

-hence we cannot derive an orthogonal basis from this transformation

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## What coordinate system is used for constant elliptical surfaces?

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-elliptic cylindrical coordinates, {u, φ, z}

x = a*cosh(u)*cosφ , y = a*sinh(u)*sinφ, z = z

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##
Elliptic Cylindrical Coordinates

Surfaces of constant u

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x = a*cosh(u)*cosφ

y = a*sinh(u)*sinφ

-rearrange

cosφ = x /a*cosh(u), sinφ = y/a*sinh(u)

sin²φ + cos²φ = 1 = x²/a²cosh²u + y²/a²sinh²u

-which for CONSTANT u is the equation of an elliptical surface with:

A = a cosh(u) and B = a sinh(u)

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##
Elliptic Cylindrical Coordinates

Surfaces of constant φ

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x = a*cosh(u)*cosφ

y = a*sinh(u)*sinφ

-rearrange

cosh(u) = x/a*cosφ , sinh(u) = y/a*sinφ

-using the identity, cosh²(u) - sinh²(u) = 1

x² / a²cos²φ - y² / a²sin²φ = 1

-therefore surfaces of constant φ are hyperbolic

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##
General Expression for Grad

Equation

### grad f = 1/hu ∂f/∂u ^eu + 1/hv ∂f/∂v ^ev + 1/hw ∂f/∂w ^ew

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##
General Expression for Grad

Derivation

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-let f be a scalar field expressed in terms of curvilinear coordinates u, v and w

-recall that the change in f due to an infinitesimal change in displacement d|x is:

df = (grad f) . d|x

-and from the complete derivative definition:

df = ∂f/∂u du + ∂f/∂v dv + ∂f/∂w dw

-in curvilinear coordinates,:

d|x = hu ^eu du + hv ^ev dv + hw ^ew dw

-if we choose a displacement such that dv=dw=0 :

df = ∂f/∂u du = (grad f ) . ^eu hu du

-this gives

(grad f) . ^eu = 1/hu ∂f/∂u

-by definition the L.H.S. is the u component of grad f

-similarly for the other two directions:

grad f = 1/hu ∂f/∂u ^eu + 1/hv ∂f/∂v ^ev + 1/hw ∂f/∂w ^ew

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## General Expression for Divergence

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div |F =

1/(huhvhw) * [ ∂/∂u(Fuhvhw) + ∂/∂v(Fvhuhw) + ∂/∂w(Fwhuhv) ]

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## Laplacian of a Scalar in Curvilinear Coordinates

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-to find ∇²f we use the identity ∇²f = div(grad f)

-sub in the formulas for grad and div in curvilinear coordinates:

∇²f = 1/huhvhw * [ ∂/∂u (hvhw/hu ∂f/∂u) + ∂/∂v (huhw/hv ∂f∂v + ∂/∂w (huhv/hw ∂f/∂w) ]

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##
Application of the Laplacian in Curvilinear Coordinates

Newton's Law of Gravity Equation

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-Newton's Law of Gravity can be written as:

|∇ . |g(|x) = - 4πG ρ(|x)

-where ρ is mass density, a scalar field

-G is the universal gravitational constant

-|g is the gravitational field

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##
Application of the Laplacian in Curvilinear Coordinates

Poisson's Equation for Gravitational Potential

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-gravitational field |g is conserevative so can be written as grad of a potential:

|g = ∇Φ

-so:

|∇ . (|∇Φ) = - 4πG ρ(|x)

using the identity ∇²f = div(grad f)

∇²Φ = - 4πG ρ(|x)

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##
Application for the Laplacian in Curvilinear Coordinates

Variation of ρ with Position

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-consider a spherical ball of mass M and radius a centred on the origin

-the mass density out side of the ball i.e. for r>a is 0

-the mass density within the ball i.e. fo r≤a is M/(4πr³/3 = 3M/4πr³

-in general we expect Φ to be independent of θ and φ, we expect spherical symmetry as ρ only depends on r

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## General Form of Curl in Curvilinear Coordinates

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|∇ x |F = 1/huhvhw * det (3x3)

where det(3x3) is the determinant of a 3x3 matrix:

-first row: hu ^eu , hv ^ev , hw ^ew

-second row: ∂/∂u , ∂/∂v , ∂/∂w

-third row: hu Fu , hv Fv , hw Fw

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