lecture12-09

lecture12-09 - Math 18.02(Spring 2009 Lecture 12 Vector...

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Unformatted text preview: Math 18.02 (Spring 2009): Lecture 12 Vector fields, Gradient, directional derivatives, tangent plane to (level) curves and surfaces March 3 Last time: Chain Rule. Differential. Today: Vector fields, Gradient, directional derivatives, tangent plane to (level) surfaces. Reading Material: From Simmons : 19.5. 2 Vector Fields In 2D a vector field is a vector valued function ~ F ( x, y ) = f ( x, y ) ˆ i + g ( x, y ) ˆ j , where f ( x, y ) and g ( x, y ) are two scalar functions representing the coordinates. To represent a vector field graphically we draw, starting at the point ( x, y ), the vector ~ F ( x, y ) = f ( x, y ) ˆ i + g ( x, y ) ˆ j . In general one needs to recognize some special features of the vector field at hand in order to represent it in a meaningful way. Exercise 1. Represent the vector fields ~ F = x ˆ i + y ˆ j and ~ G =- y ˆ i + x ˆ j 1 In 3D a vector field is a vector valued function ~ F ( x, y, z ) = f ( x, y, z ) ˆ i + g ( x, y, z ) ˆ j + h ( x, y, z ) ˆ k , where f ( x, y, z ) , g ( x, y, z ) and h ( x, y, z ) are three scalar functions representing the coordinates. Exercise 2. Represent the vector field ~ F ( x, y, z ) = x ˆ i + y ˆ j + x ˆ k . Definition 1 (Gradient) . Given a scalar function f ( x, y, z ) we can define a very special vector field linked to f called gradient : ~ O f ( x, y, z ) = f x ( x, y, z ) ˆ i + f y ( x, y, z ) ˆ j + f z ( x, y, z ) ˆ k The symbol ~ O formally can be defined as ~ O = ˆ i∂ x + ˆ j∂ y + ˆ k∂ z ....
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lecture12-09 - Math 18.02(Spring 2009 Lecture 12 Vector...

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