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Path Independence for Line Integrals

By Salman Khan - Khan Academy
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Lecture Description

Showing that if a vector field is the gradient of a scalar field, then its line integral is path independent.

Course Description

Topics covered in the first two or three semesters of college calculus. Everything from limits to derivatives to integrals to vector calculus. Should understand the topics in the pre-calculus playlist first (the limit videos are in both playlists).

Course Index

  1. Epsilon Delta Limit Definition Part 1
  2. Epsilon Delta Limit Definition Part 2
  3. Calculus: Derivatives 1
  4. Calculus: Derivatives 2
  5. Calculus: Derivatives 2.5
  6. Derivatives Part 1
  7. Derivatives Part 2
  8. Derivatives Part 3
  9. Derivatives Part 4
  10. Derivatives Part 5
  11. Derivatives Part 6
  12. Derivatives Part 7
  13. Derivatives Part 8
  14. Derivatives Part 9
  15. Proof: d/dx(x^n)
  16. Proof: d/dx(sqrt(x))
  17. Proof: d/dx(ln x) = 1/x
  18. Proof: d/dx(e^x) = e^x
  19. Proofs of Derivatives of Ln(x) and e^x
  20. Extreme Derivative Word Problem
  21. Implicit Differentiation Part 1
  22. Implicit Differentiation Part 2
  23. More Implicit Differentiation
  24. More Chain Rule and Implicit Differentiation Intuition
  25. Trig Implicit Differentiation Example
  26. Derivative of x^(x^x)
  27. Maxima Minima Slope Intuition
  28. Inflection Points and Concavity Intuition
  29. Monotonicity Theorem
  30. Maximum and Minimum Values on an Interval
  31. Graphing Using Derivatives
  32. Graphing with Derivatives Example
  33. Graphing with Calculus
  34. Optimization with Calculus Part 1
  35. Optimization with Calculus Part 2
  36. Optimization with Calculus Part 3
  37. Optimization with Calculus Part 4
  38. Introduction to Rate-of-change
  39. Equation of a Tangent Line
  40. Rates-of-change Part 2
  41. Ladder rate-of-change Problem
  42. Mean Value Theorem
  43. Indefinite Integrals Part 1
  44. Indefinite Integrals Part 2
  45. Indefinite Integrals Part 3
  46. Indefinite Integrals Part 4
  47. Indefinite Integrals Part 5
  48. Indefinite Integrals Part 6
  49. Indefinite Integrals Part 7
  50. Another U-substitution Example
  51. Definite Integrals Part 1
  52. Definite Integrals Part 2
  53. Definite Integrals Part 3
  54. Definite Integrals Part 4
  55. Definite Integrals Part 5
  56. Definite Integral with Substitution
  57. Integrals: Trig Substitution Part 1
  58. Integrals: Trig Substitution Part 2
  59. Integrals: Trig Substitution Part 3
  60. Introduction to Differential Equations
  61. Solid of Revolution Part 1
  62. Solid of Revolution Part 2
  63. Solid of Revolution Part 3
  64. Solid of Revolution Part 4
  65. Solid of Revolution Part 5
  66. Solid of Revolution Part 6
  67. Solid of Revolution Part 7
  68. Solid of Revolution Part 8
  69. Polynomial Approximation of Functions Part 1
  70. Polynomial Approximation of Functions Part 2
  71. Polynomial Approximation of Functions Part 3
  72. Polynomial Approximation of Functions Part 4
  73. Polynomial Approximation of Functions Part 5
  74. Polynomial Approximation of Functions Part 6
  75. Polynomial Approximation of Functions Part 7
  76. Taylor Polynomials
  77. AP Calculus BC Exams: 2008 1 A
  78. AP Calculus BC Exams: 2008 1 B & C
  79. AP Calculus BC Exams: 2008 1 C & D
  80. AP Calculus BC Exams: 2008 1 D
  81. Calculus BC 2008 2 A
  82. Calculus BC 2008 2 B & C
  83. Calculus BC 2008 2 D
  84. Partial Derivatives Part 1
  85. Partial Derivatives Part 2
  86. Gradient
  87. Gradient of a Scalar Field
  88. Divergence Part 1
  89. Divergence Part 2
  90. Divergence Part 3
  91. Curl Part 1
  92. Curl Part 2
  93. Curl Part 3
  94. Double Integrals Part 1
  95. Double Integrals Part 2
  96. Double Integrals Part 3
  97. Double Integrals Part 4
  98. Double Integrals Part 5
  99. Double Integrals Part 6
  100. Triple Integrals Part 1
  101. Triple Integrals Part 2
  102. Triple Integrals Part 3
  103. (2^ln x)/x Antiderivative Example
  104. Line Integrals
  105. Line Integral Example 1
  106. Line Integral Example 2 Part 1
  107. Line Integral Example 2 Part 2
  108. Position Vector Valued Functions
  109. Derivative of a Position Vector Valued Function
  110. Differential of a Vector Valued Function
  111. Differential of a Vector Valued Function Example
  112. Line Integrals and Vector Fields
  113. Using a Line Integral to Find a Vector Field Example
  114. Parametrization of a Reverse Path
  115. Scalar Field Line Integral Independent of Path Direction
  116. Vector Field Line Integral Dependent of Path Direction
  117. Path Independence for Line Integrals
  118. Closed Curve Line Integrals of Conservative Vector Fields
  119. Example of Closed Line Integral of Conservative Field
  120. Second Example of Line Integral of Conservative Vector Field
  121. Green's Theorem Proof Part 1
  122. Green's Theorem Proof Part 2
  123. Green's Theorem Example Part 1
  124. Green's Theorem Example Part 2
  125. Introduction to Parametrizing a Surface with Two Parameters
  126. Position Vector-Valued Function for a Parametrization of Two Parameters
  127. Partial Derivatives of Vector-Valued Functions
  128. Introduction to the Surface Integral
  129. Calculating a Surface Integral Example Part 1
  130. Calculating a Surface Integral Example Part 1
  131. Calculating a Surface Integral Example Part 1
  132. L'Hopital's Rule
  133. L'Hopital's Rule Example 1
  134. L'Hopital's Rule Example 2
  135. L'Hopital's Rule Example 3