Concrete example of the derivative of a vector valued function to better understand what it means

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Multivariable Calculus on Khan Academy: Think calculus. Then think algebra II and working with two variables in a single equation. Now generalize and combine these two mathematical concepts, and you begin to see some of what Multivariable calculus entails, only now include multi dimensional thinking. Typical concepts or operations may include: limits and continuity, partial differentiation, multiple integration, scalar functions, and fundamental theorem of calculus in multiple dimensions.





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1. Multivariable functions
2. Representing points in 3d
3. Introduction to 3d graphs
4. Interpreting graphs with slices
5. Contour plots
6. Parametric curves
7. Parametric surfaces
8. Vector fields, introduction
9. Fluid flow and vector fields
10. 3d vector fields, introduction
11. 3d vector field example
12. Transformations, part 1
13. Transformations, part 2
14. Transformations, part 3
15. Partial derivatives, introduction
16. Partial derivatives and graphs
17. Formal definition of partial derivatives
18. Symmetry of second partial derivatives
19. Gradient
20. Gradient and graphs
21. Directional derivative
22. Directional derivative, formal definition
23. Directional derivatives and slope
24. Why the gradient is the direction of steepest ascent
25. Gradient and contour maps
26. Position vector valued functions | Multivariable Calculus | Khan Academy
27. Derivative of a position vector valued function | Multivariable Calculus | Khan Academy
28. Differential of a vector valued function | Multivariable Calculus | Khan Academy
29. Vector valued function derivative example | Multivariable Calculus | Khan Academy
30. Multivariable chain rule
31. Multivariable chain rule intuition
32. Vector form of the multivariable chain rule
33. Multivariable chain rule and directional derivatives
34. More formal treatment of multivariable chain rule
35. Curvature intuition
36. Curvature formula, part 1
37. Curvature formula, part 2
38. Curvature formula, part 3
39. Curvature formula, part 4
40. Curvature formula, part 5
41. Curvature of a helix, part 1
42. Curvature of a helix, part 2
43. Curvature of a cycloid
44. Computing the partial derivative of a vector-valued function
45. Partial derivative of a parametric surface, part 1
46. Partial derivative of a parametric surface, part 2
47. Partial derivatives of vector fields
48. Partial derivatives of vector fields, component by component
49. Divergence intuition, part 1
50. Divergence intuition, part 2
51. Divergence formula, part 1
52. Divergence formula, part 2
53. Divergence example
54. Divergence notation
55. 2d curl intuition
56. 2d curl formula
57. 2d curl example
58. 2d curl nuance
59. Describing rotation in 3d with a vector
60. 3d curl intuition, part 1
61. 3d curl intuition, part 2
62. 3d curl formula, part 1
63. 3d curl formula, part 2
64. 3d curl computation example
65. Laplacian intuition
66. Laplacian computation example
67. Explicit Laplacian formula
68. Harmonic Functions
69. What is a tangent plane
70. Controlling a plane in space
71. Computing a tangent plane
72. Local linearization
73. What do quadratic approximations look like
74. Quadratic approximation formula, part 1
75. Quadratic approximation formula, part 2
76. Quadratic approximation example
77. The Hessian matrix
78. Expressing a quadratic form with a matrix
79. Vector form of multivariable quadratic approximation
80. Multivariable maxima and minima
81. Saddle points
82. Warm up to the second partial derivative test
83. Second partial derivative test
84. Second partial derivative test intuition
85. Second partial derivative test example, part 1
86. Second partial derivative test example, part 2

In multivariable calculus, we progress from working with numbers on a line to points in space. It gives us the tools to break free from the constraints of one-dimension, using functions to describe space, and space to describe functions.

The only thing separating multivariable calculus from ordinary calculus is this newfangled word "multivariable". It means we will deal with functions whose inputs or outputs live in two or more dimensions. Here we lay the foundations for thinking about and visualizing multivariable functions.
Introduction to multivariable calculus