lecture29-09

# lecture29-09 - 18.02 Multivariable Calculus(Spring 2009...

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Unformatted text preview: 18.02 Multivariable Calculus (Spring 2009): Lecture 29 Cylindrical and Spherical Coordinates and applications April 17 Reading Material: From Simmons : 18.7, 20.6 and 20.7. Last time: Triple Integrals. Cylindrical Coordinates. Today: Cylindrical and Spherical Coordinates. Applications. 2 Cylindrical Coordinates Cylindrical coordinates are obtained by using polar coordinates in the xy plane and the z coordinate along the z-axis, to summarize: rectangular cylindrical = polar + z ( x, y, z ) ( r, θ, z ) Conversion: Like polar: ( r, θ, z )-→ ( x, y, z ) x = r cos θ y = r sin θ z = z ( r, θ, z ) ←- ( x, y, z ) r = p x 2 + y 2 θ = tan- 1 y/x z = z Integration in Cylindrical Coordinates: ZZZ D f ( x, y, z ) dV-→ ZZZ θ r z f ( r cos x, r sin x, z ) dV z }| { r dz dr dθ limits 1 Question: Why dV = r dz dr dθ ? (2.1) If one wiggles Δ r , Δ θ , Δ z how big is resulting ”cube” ? From the picture is then clear that Δ V ≈ ( r Δ θ )(Δ r )(Δ z ) hence if we pass to differentials we have dV = r dz dr dθ | {z } typical order which makes sense since we have to take into account the rate of change of the area of the base (with respect to θ and r ) and the rate of change of the hight with respect to...
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lecture29-09 - 18.02 Multivariable Calculus(Spring 2009...

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