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Engineering Calculus Notes 319

Engineering Calculus Notes 319 - 3.5 SURFACES AND THEIR...

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Unformatted text preview: 3.5. SURFACES AND THEIR TANGENT PLANES 307 and ﬁnding lines tangent to the resulting curves. A more fruitful view, however, is to think in terms of arbitrary curves in the surface. Suppose → − (r, s) is a C 1 function parametrizing the surface S in R3 and p → → P = − (r0 , s0 ) is a regular point; by restricting the domain of − we can p p assume that we have a coordinate patch for S. Any curve in S can be represented as → − (t) = − (r (t) , s(t)) → γ p or x = x(r (t) , s(t)) y = y (r (t) , s(t)) z = z (r (t) , s(t)) —that is, we can “pull back” the curve on S to a curve in the parameter space. If we want the curve to pass through P when t = 0, we need to require r (0) = r0 s(0) = s0 . If r (t) and s(t) are diﬀerentiable, then by the Chain Rule γ (t) is also diﬀerentiable, and its velocity vector can be found via → → − (t) = − (t) ˙ γ v = dx dx dx ,, dt dt dt where ∂x dr ∂x ds dx = + dt ∂r dt ∂s dt dy ∂y dr ∂y ds = + dt ∂r dt ∂s dt dz ∂z dr ∂z ds = + . dt ∂r dt ∂s dt → We expect that for any such curve, − (0) will be parallel to the tangent v plane to S at P . In particular, the two curves obtained by holding one of ...
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