5-Tang Plain Chain Rule

5-Tang Plain Chain Rule - Tangent Plane Chain Rule John E...

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Unformatted text preview: Tangent Plane, Chain Rule John E. Gilbert, Heather Van Ligten, and Benni Goetz Start with a function z = f ( x, y ) and let P = ( a, b, f ( a, b )) be the point on the graph of z = f ( x, y ) above a point ( a, b ) in the xy-plane as shown to the right below . Slicing the surface with vertical planes y = b and x = a creates two curves on this graph both passing through P . These two space curves are the graphs of vector functions r 1 ( x ) = x, b, f ( x, b ) , r 2 ( y ) = a, y, f ( a, y ) , shown in orange on the surface. The vector derivatives T 1 = r 1 ( a ) = 1 , , f x ( a, b ) = i + f x ( a, b ) k , T 2 = r 2 ( b ) = , 1 , f y ( a, b ) = j + f y ( a, b ) k , are Tangent Vectors at P to the graph of f , while the plane containing P as well as T 1 and T 2 is the Tangent Plane at P . To calculate the equation of the tangent plane at P we need its normal: n = T 1 × T 2 = i j k 1 f x ( a, b ) 1 f y ( a, b ) =- f x ( a, b ) ,- f y ( a, b ) , 1 . On the other hand, if Q ( x, y, z ) is an arbitrary point in the tangent plane at P , then-→ PQ = x- a, y- b, z- f ( a, b ) lies in the plane and so is perpendicular to n . Thus in point-normal form the equation of the tangent plane is x- a, y- b, z- f ( a, b ) · n = x- a, y- b, z- f ( a, b ) · - f x ( a, b ) ,- f y ( a, b ) , 1 =- ( x- a ) f x ( a, b )- ( y- b ) f y ( a, b ) + z- f ( a, b ) = 0 . After rearranging we get: The equation of the Tangent Plane at ( a, b, f ( a, b )) is z = f ( a, b ) + ( x- a ) f x ( a, b ) + ( y- b ) f y ( a, b ) , while the Linearization of f at ( a, b ) is L ( x, y ) = f ( a, b ) + ( x- a ) f x ( a, b ) + ( y- b ) f y ( a, b ) ....
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5-Tang Plain Chain Rule - Tangent Plane Chain Rule John E...

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