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Unformatted text preview: s the direction vector for the tangent line to the level curve. is Normal to the Tangent Line of Level Curve Later in the book, we will do an application where we want to find the curves Perpendicular to the Level Curves.
Find all curves perpendicular to the level curves of the surface
So we have for our level curves. (circles) Let be a parameterization of the level curve. Let our curve that is perpendicular to this curve be represented by . We want: We know Thus
So we want and These are First Order (Separable) Differential Equations They are easy to solve: The same logic would hold for the other equation to give us:
Thus (Lines that go through the origin) Definition of Directional Derivative:
Let be a unit vector. The directional derivative of f in the direction of is given by: Show that the maximum rate of change of f is in the direction of the Gradient
We know from chapter 10: This value will be maximized when . When When and point in the same direction. . When When and point in the opposite direction. The maximum value will be:
The value will be minimized when
The minimum value will be: The Path of Steepest Descent or Ascent: A curve that is perpendicular to each level curve through which it passes
This path is dependent upon starting point and step size taken for each direction
The temperature distribution across the surface of a rectangular plate is given by the function
Find the path followed by a heat-seeking particle placed on the plate at the point ( 4 , 2 ).
We will parameterize the path with the vector function
We want the tangent vector to point in the direction of greatest temperature change ( ie: the gradient
Thus we want
We can solve these differential equations by separation of variables to get
Thus the path is:
The Directional Derivative
The directional derivative is a generalization of the partial derivatives.
means the rate of change PARALLEL to the x-axis
means the rate of change PARALLEL to the y-axis. ) The definition of the partial derivatives: We can write these in vector form. Let Instead of
derivative: Notice if
Also if or Replace these with an arbitrary unit...
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This note was uploaded on 03/05/2014 for the course MATH 101c taught by Professor Loukianoff,v during the Spring '08 term at Ohlone.
- Spring '08