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Orthogonal Curvilinear Coordinates

# Orthogonal Curvilinear Coordinates - Chapter 5 Orthogonal...

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Unformatted text preview: Chapter 5 Orthogonal Curvilinear Coordinates Syllabus section: 4. Orthogonal curvilinear coordinates; length of line element; grad, div and curl in curvilinear coordinates; spherical and cylindrical polar coordinates as examples. So far we have only used Cartesian coordinates. Often, because of the geometry of the problem, it is easier to work in some other coordinate system. Here we show how to do this, restricting the generality only by an orthogonality condition. 5.1 Plane Polar Coordinates In Calculus II you met the simple curvilinear coordinates in two dimensions, plane polars, defined by x = r cos θ , y = r sin θ . We can easily invert these relations to get r = radicalbig x 2 + y 2 , θ = arctan ( y / x ) . The Chain Rule enables us to relate partial derivatives with respect to x and y to those with respect to r and θ and vice versa, e.g. ∂ f ∂ r = ∂ x ∂ r ∂ f ∂ x + ∂ y ∂ r ∂ f ∂ y = x r ∂ f ∂ x + y r ∂ f ∂ y . (5.1) In Calculus II, the rule for changing coordinates in integrals is also given. The general rule is that if x = x ( u , v ) , y = y ( u , v ) then the d x d y in the integral is replaced by vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle ∂ x ∂ u ∂ x ∂ v ∂ y ∂ u ∂ y ∂ v vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle d u d v . This is just d S = | r u × r v | d u d v , as derived in section 3.2. For plane polar coordinates this gives just d S = r d r d θ . 49 Example 5.1. The Gaussian integral (related to the Gaussian distribution in statistics) Consider the integral integraldisplay ∞ − ∞ integraldisplay ∞ − ∞ e − ( x 2 + y 2 ) dxdy = parenleftBig integraltext ∞ − ∞ e − x 2 dx parenrightBig 2 . Transforming to polar coordinates gives integraldisplay ∞ re − r 2 d r integraldisplay 2 π d θ = [ − 1 2 e − r 2 ] ∞ [ θ ] 2 π = π and hence (according to Dr. Saha “the most beautiful of all integrals”) integraldisplay ∞ − ∞ e − x 2 d x = √ π . For later use, consider the unit vectors in the directions in which r and θ increase at a point, which we denote e r and e θ . These point along the coordinate lines, a coordinate line meaning a curve on which only one of the coordinates is varying. Coordinate lines are generalizations of coordinate axes. We know how to find the displacements arising from small changes in the coordinates, so all we have to do is divide the displacements by their lengths. Thus d r r = r r d r = (( cos θ ) i + ( sin θ ) j ) d r , | r r | = 1 ⇒ e r = r r = cos θ i + sin θ j while d r θ = r θ d θ = (( − r sin θ ) i + ( r cos θ ) j ) d θ , | r θ | = r ⇒ e θ = r θ / r = − sin θ i + cos θ j ....
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Orthogonal Curvilinear Coordinates - Chapter 5 Orthogonal...

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