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Unformatted text preview: Section 14.6 Tangent Planes and Differentials Shubhodeep Mukherji In this section we define the tangent plane at a point on a smooth surface in space. We calculate an equation of the tangent plane from the partial derivatives of the function defining the surface. We will then study the total differential and linearization of functions of several variables. 1 Tangent Planes and Normal Lines If r = g ( t ) i + h ( t ) j + k ( t ) k is a smooth curve on the level surface f ( x,y,z ) = c of a differentiable function f , then f ( g ( t ) ,h ( t ) ,k ( t )) = c . Differentiating both sides wrt t leads to d dt f ( g ( t ) ,h ( t ) ,k ( t )) = d dt ( c ) ∂f ∂x dg dt + ∂f ∂y dh dt + ∂f ∂z dk dt = 0 ∂f ∂x i + ∂f ∂y j + ∂f ∂z k dg dt i + dh dt j + dk dt k = 0 ( ∇ f ) d r dt = 0 At every point along the curve, ∇ f is orthogonal to the curve’s velocity vector. 1.1 Tangent Plane The tangent plane at the point P ( x ,y ,z ) on the level surface f ( x,y,z ) = c of a differentiable function f is the plane throught P normal to ∇ f  P . The equation for the tangent plane to f ( x,y,z ) = c at P ( x ,y ,z ) is f x ( P )( x x ) + f y ( P )( y y ) + f z ( P )( z z ) = 0 1 1.2 Normal Line The normal line of the surface at P is the line through P parallel to ∇ f  P . The equation for the normal line to f ( x,y,z ) = c at P ( x ,y ,z ) is x = x + f x ( P ) t,y = y + f y ( P ) t,z = z + f z ( P ) t Example 1 Find the tangent plane and normal line of the surface f ( x,y,z ) = x 2 + y 2 + z 9 = 0 at the point P (1 , 2 , 4) . The tangent plane is the plane through P perpendicular to the gra dient of f at P . The gradient is ∇ f P = (2 x i + 2 y j + k ) (1 , 2 , 4) = 2 i + 4 j + k The tangent plane is therefore the plane 2( x 1) + 4( y 2) + ( z 4) = 0 2 x + 4 y + z = 14 The line normal to the surface at P is x = 1 + 2 t y = 2 + 4 t z = 4 + t 1.3 Plane Tangent to a Surface To find an equation for the plane tangent to a smooth surface z = f ( x,y ) at a point P ( x ,y ,z ) where z = f ( x ,y ), we first observe that the equation z = f ( x,y ) is equivalent to f ( x,y ) z = 0. The surface z = f ( x,y ) is therefore the zero level surface of the function F ( x,y,z ) = f ( x,y ) z . The partial derivatives of F are F x = ∂ ∂x ( f ( x,y ) z ) = f x 0 = f x F y = ∂ ∂y ( f ( x,y ) z ) = f y 0 = f y 2 F z = ∂ ∂z ( f ( x,y ) z ) = 0 1 = 1 The formula F x ( P )( x x ) + F y ( P )( y y ) + F z ( P )( z z ) = 0 is for the plane tangent to the level surface at P therefore reduces to f x ( x ,y )( x x ) + f y ( x ,y )( y y ) ( z z ) = 0 The above equation is for the plane tangent to a surface z = f ( x,y ) of a differentiable function f at the point P ( x ,y ,z ) = ( x ,y ,f ( x ,y 0))....
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 Spring '11
 profeessor
 Numerical Analysis, Derivative, Differential Calculus, Jacobian matrix and determinant

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