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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 P_{0}(x_{0}, y_{0}, z_{0}) of the curve. (computed at t = t_{0})

Equation of normal plane at point P_{0}(x_{0}, y_{0}, z_{0}) of the curve. (normal plane is plane through
P_{0} perpendicular to the tangent line)

It is understood that the derivatives in both 1] and 2] are evaluated at point P_{0}.

Surfaces.

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I. Let a surface be defined by the equation z = f(x, y). Let P_{0}(x_{0}, y_{0}, z_{0}) be any point on it.

Equation of the tangent plane at point P_{0}. See Fig. 1.

where the partials are evaluated at point P_{0}.

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 P_{0}. (The normal line is the line perpendicular to the
tangent plane at point P_{0}.)

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II. Let a surface be defined by the equation F(x, y, z) = 0. Let P_{0}(x_{0}, y_{0}, z_{0}) be any point on it.

Equation of the tangent plane at point P_{0}.

Equation of the normal line at point P_{0}.

where it is understood that all partial derivatives are evaluated at point P_{0}.

References.

Ayres. Calculus. (Schaum)

Middlemiss. Differential and Integral Calculus.

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