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SPACE CURVES AND SURFACES

Space curves. Let a space curve be defined parametrically by the equations

x = f(t)

y = g(t)

z = h(t) .

Equation of tangent line at point P0(x0, y0, z0) of the curve. (computed at t = t0)

Equation of normal plane at point P0(x0, y0, z0) of the curve. (normal plane is plane through P0 perpendicular to the tangent line)

It is understood that the derivatives in both 1] and 2] are evaluated at point P0.

Surfaces.

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I. Let a surface be defined by the equation z = f(x, y). Let P0(x0, y0, z0) be any point on it.

Equation of the tangent plane at point P0. See Fig. 1.

where the partials are evaluated at point P0.

Theorem. The normal to a surface z = f(x, y) at a point P has direction numbers

where the partials are evaluated at point P.

Equation of the normal line at point P0. (The normal line is the line perpendicular to the tangent plane at point P0.)

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II. Let a surface be defined by the equation F(x, y, z) = 0. Let P0(x0, y0, z0) be any point on it.

Equation of the tangent plane at point P0.

Equation of the normal line at point P0.

where it is understood that all partial derivatives are evaluated at point P0.

References.

Ayres. Calculus. (Schaum)

Middlemiss. Differential and Integral Calculus.