1. Chapter 14 – Partial Derivatives
14.6 Directional Derivatives and the Gradient Vector
Objectives:
Determine the directional derivative of a vector
Determine the gradient of a vector
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14.6 Directional Derivatives and the Gradient Vector

2. Definition – Directional Derivative
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14.6 Directional Derivatives and the Gradient Vector

3. Visualization Directional Derivatives
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14.6 Directional Derivatives and the Gradient Vector

4. Theorem 14.6 Directional Derivatives and the Gradient Vector
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14.6 Directional Derivatives and the Gradient Vector

5. Note
If the unit vector u makes an angle  with the positive x-axis, then we can write u = <cos , sin  > and the formula in theorem 3 becomes
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14.6 Directional Derivatives and the Gradient Vector

6. Example 1- pg. 920 # 4
Find the directional derivative of f at the given point in the direction indicated by the angle θ.
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14.6 Directional Derivatives and the Gradient Vector

7. Example 2
Find the directional derivative of f at the given point in the direction indicated by the angle θ.
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14.6 Directional Derivatives and the Gradient Vector

8. Definition - Gradient The gradient of f, ∇f, is read “del f”.
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14.6 Directional Derivatives and the Gradient Vector

9. Properties of the Gradient
The gradient of a multi-variable function has a component for each direction.
Be careful not to confuse the coordinates and the gradient. The coordinates are the current location, measured on the x-y-z axis.
The gradient is a direction to move from our current location, such as move up, down, left or right.
The gradient at any location points in the direction of greatest increase of a function.
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14.6 Directional Derivatives and the Gradient Vector

10. Directional Derivative and Gradient
We can now rewrite the directional derivative as
which expresses the directional derivative in the direction of u as the scalar projection of the gradient vector onto u.
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14.6 Directional Derivatives and the Gradient Vector

11. Directional Derivative and Gradient
What is the difference between the two?
The gradient is a vector quantity that points in the direction of greatest increase and has a magnitude equal to the rate of change in that direction. 
The directional derivative is a scalar quantity giving the rate of change in a specified direction. It is equal to the dot product of the gradient with a unit vector in the desired direction. 
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14.6 Directional Derivatives and the Gradient Vector

12. Directional Derivative and Gradient
What is the difference between the two?
The gradient uses all of the partials.
A directional derivative uses only one partial.
The gradient is a vector orthogonal to a level curve (surface, etc.).
A directional derivative measures how function values change with respect to movement in a particular direction in the domain...and it is not a vector 
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14.6 Directional Derivatives and the Gradient Vector

13. Example 3 – pg. 943 # 8 Find the gradient of f.
Evaluate the gradient at the point P.
Find the rate of change of f at P in the direction of vector u.
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14.6 Directional Derivatives and the Gradient Vector

14. Definition – Directional Derivative
If we use vector notation, then the definition is more compact.
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14.6 Directional Derivatives and the Gradient Vector

15. Definition - Gradient The gradient of f is and can be rewritten as
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14.6 Directional Derivatives and the Gradient Vector

16. Example 4 Find the gradient of f.
Evaluate the gradient at the point P.
Find the rate of change of f at P in the direction of vector u.
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14.6 Directional Derivatives and the Gradient Vector

17. Maximizing the Directional Derivative
Visualization
Maximizing the Directional Derivative
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14.6 Directional Derivatives and the Gradient Vector

18. Example 5
Find the maximum rate of change of f at the given point and the direction in which it occurs.
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14.6 Directional Derivatives and the Gradient Vector

19. Example 6
Find the maximum rate of change of f at the given point and the direction in which it occurs.
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14.6 Directional Derivatives and the Gradient Vector

20. The equation ∇F(x0,y0,z0)∙r’(t0) says that the gradient vector at P, ∇F(x0,y0,z0),is perpendicular to the tangent vector r’(t0) to any curve C on S that passes through P.
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14.6 Directional Derivatives and the Gradient Vector

21. Definition
If ∇F(x0,y0,z0)≠0, it is thus natural to define the tangent plane to the level surface F(x, y, z) = k at P(x0, y0, z0) as the plane that passes through P and has normal vector ∇F(x0,y0,z0). This equation of a tangent plane can be written as
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14.6 Directional Derivatives and the Gradient Vector

22. Normal Line The normal line to S at P is the line: Passing through P
Perpendicular to the tangent plane
The direction of the normal line is given by the gradient vector
∇F(x0,y0,z0)
and its symmetric equations are
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14.6 Directional Derivatives and the Gradient Vector

23. Example 7
Find the equations of (a) the tangent plane and (b) the normal line to the given surface at the specified point.
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14.6 Directional Derivatives and the Gradient Vector

24. Group work
1. Pg. 944 # 28
Find the directions in which the directional derivative of the function below at the point (0,2) has the value of 1.
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14.6 Directional Derivatives and the Gradient Vector

25. Group work
2. Pg. 944 # 32
The temperature at a point (x, y, z) is given by
Where T is measured in oC and x, y, z in meters.
Find the rate of change of temperature at the point P(2,-1,2) in the direction towards the point (3, -3, 3).
In which direction does the temperature increase the fastest at P?
Find the maximum rate of increase at P.
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14.6 Directional Derivatives and the Gradient Vector

26. Group work
3. Pg. 944 # 34
Suppose you are climbing a hill whose shape is given by the equation , where x, y, and z are measured in meters, and you are standing at the point with coordinates (60, 40, 996). The positive x-axis points east and the positive y-axis points another.
If you walk due south, will you start to ascend or descend? At what rate?
If you walk northwest, will you start to ascend or descend? At what rate?
In which direction is the slope the largest? What is the rate of ascent in that direction? At what angle above the horizontal does the path in that direction begin?
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14.6 Directional Derivatives and the Gradient Vector

27. Group work 4. Pg. 945 # 50 If find the gradient vector
and use it to find the tangent line to the level curve g(x, y)=1 at the point (1,2). Sketch the level curve, the tangent line, and the gradient vector.
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14.6 Directional Derivatives and the Gradient Vector

28. More Examples
The video examples below are from section in your textbook. Please watch them on your own time for extra instruction. Each video is about 2 minutes in length.
Example 4
Example 5
Example 8
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14.6 Directional Derivatives and the Gradient Vector

29. Demonstrations Feel free to explore these demonstrations below.
Directional Derivatives in 3D
Maximizing Directional Derivatives
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14.6 Directional Derivatives and the Gradient Vector