1. VC.03 Gradient Vectors, Level Curves, Maximums/Minimums/Saddle Points

2. Example 1: The Gradient Vector

4. Example 2: The Gradient Vector

5. Definition: The Gradient Vector

7. Example 3: A Surface and Gradient Field

10. Example 4: A Surface and Contour Plot

11. Example 4: Contour Plots in Real Life

12. Example 4: A Surface and Contour Plot

14. Example 5: The Gradient Points in the Direction of Greatest Initial Increase

15. Example 6: A Path Along our Surface

17. Example 6: Tangent Vectors and Gradient Vectors on Our Path

19. The Derivative of f(x(t),y(t)) With Respect to t

20. The Chain Rule in n-Dimensions

21. Chain Rule Proves the Gradient is Perpendicular to the Level Curve

22. Example 7: Using the Chain Rule

23. Example 8: Identifying Local Extrema from the Gradient

28. Example 9: Another Visit to Our Surface

29. Example 10: Level Surfaces and Entering the Fourth Dimension

31. Example 11: A Different Analogy for the Fourth Dimension?

33. z = f(x,y)w = f(x,y,z) Graph3-D surface4-D (hyper)surface Can’t truly graph it! Level SetsLevel Curve k = f(x,y) Level Surface k = f(x,y,z) Gradient Vectors2D Vectors3D Vectors Final Thoughts: f(x,y) versus f(x,y,z)