Calculus

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).

Lectures

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   Lecture 1 - Epsilon Delta Limit Definition Part 1

   Introduction to the Epsilon Delta Definition of a Limit.

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   Lecture 2 - Epsilon Delta Limit Definition Part 2

   Using the epsilon delta definition to prove a limit.

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   Lecture 3 - Calculus: Derivatives 1

   Calculus-Derivative: Understanding that the derivative is just the slope of a curve at a point (or the slope of the tangent line).

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   Lecture 4 - Calculus: Derivatives 2

   Calculus-Derivative: Finding the slope (or derivative) of a curve at a particular point.

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   Lecture 5 - Calculus: Derivatives 2.5

   Calculus-Derivative: Finding the derivative of y=x^2

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   Lecture 6 - Derivatives Part 1

   Finding the slope of a tangent line to a curve (the derivative). Introduction to Calculus.

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   Lecture 7 - Derivatives Part 2

   More intuition of what a derivative is. Using the derivative to find the slope at any point along f(x)=x^2.

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   Lecture 8 - Derivatives Part 3

   Determining the derivatives of simple polynomials.

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   Lecture 9 - Derivatives Part 4

   Part 4 of derivatives. Introduction to the chain rule.

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    Lecture 10 - Derivatives Part 5

    Examples using the Chain Rule.

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    Lecture 11 - Derivatives Part 6

    Even more examples using the chain rule.

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    Lecture 12 - Derivatives Part 7

    The product rule. Examples using the Product and Chain rules.

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    Lecture 13 - Derivatives Part 8

    Why the quotient rule is the same thing as the product rule. Introduction to the derivative of e^x, ln x, sin x, cos x, and tan x.

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    Lecture 14 - Derivatives Part 9

    More examples of taking derivatives.

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    Lecture 15 - Proof: d/dx(x^n)

    Proof that d/dx(x^n) = n*x^(n-1).

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    Lecture 16 - Proof: d/dx(sqrt(x))

    Proof that d/dx (x^.5) = .5x^(-.5).

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    Lecture 17 - Proof: d/dx(ln x) = 1/x

    Taking the derivative of ln x.

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    Lecture 18 - Proof: d/dx(e^x) = e^x

    Proof that the derivative of e^x is e^x.

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    Lecture 19 - Proofs of Derivatives of Ln(x) and e^x

    Doing both proofs in the same video to clarify any misconceptions that the original proof was "circular".

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    Lecture 20 - Extreme Derivative Word Problem

    A difficult but interesting derivative word problem.

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    Lecture 21 - Implicit Differentiation Part 1

    Taking the derivative when y is defined implicitly.

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    Lecture 22 - Implicit Differentiation Part 2

    A hairier implicit differentiation problem.

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    Lecture 23 - More Implicit Differentiation

    2 more implicit differentiation examples.

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    Lecture 24 - More Chain Rule and Implicit Differentiation Intuition

    More intuition behind the chain rule and how it applies to implicit differentiation.

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    Lecture 25 - Trig Implicit Differentiation Example

    Implicit differentiation example that involves the tangent function.

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    Lecture 26 - Derivative of x^(x^x)

    Calculus: Derivative of x^(x^x).

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    Lecture 27 - Maxima Minima Slope Intuition

    Intuition on what happens to the slope/derivative and second derivatives at local maxima and minima.

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    Lecture 28 - Inflection Points and Concavity Intuition

    Understanding concave upwards and downwards portions of graphs and the relation to the derivative. Inflection point intuition.

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    Lecture 29 - Monotonicity Theorem

    Using the monotonicity theorem to determine when a function is increasing or decreasing.

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    Lecture 30 - Maximum and Minimum Values on an Interval

    2 examples of finding the maximum and minimum points on an interval.

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    Lecture 31 - Graphing Using Derivatives

    Graphing functions using derivatives.

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    Lecture 32 - Graphing with Derivatives Example

    Using the first and second derivatives to identify critical points and inflection points and to graph the function.

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    Lecture 33 - Graphing with Calculus

    More graphing with calculus.

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    Lecture 34 - Optimization with Calculus Part 1

    Find two numbers whose products is -16 and the sum of whose squares is a minimum.

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    Lecture 35 - Optimization with Calculus Part 2

    Find the volume of the largest open box that can be made from a piece of cardboard 24 inches square by cutting equal squares from the corners and turning up the sides.

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    Lecture 36 - Optimization with Calculus Part 3

    A wire of length 100 centimeters is cut into two pieces; one is bent to form a square, and the other is bent to form an equilateral triangle. Where should the cut be made if (a) the sum of the two areas is to be a minimum; (b) a maximum? (Allow the possibility of no cut.).

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    Lecture 37 - Optimization with Calculus Part 4

    Minimizing the cost of material for an open rectangular box.

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    Lecture 38 - Introduction to Rate-of-change

    Using derivatives to solve rate-of-change problems.

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    Lecture 39 - Equation of a Tangent Line

    Finding the equation of the line tangent to f(x)=xe^x when x=1.

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    Lecture 40 - Rates-of-change Part 2

    Another (simpler) example of using the chain rule to determine rates-of-change.

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    Lecture 41 - Ladder rate-of-change Problem

    The classic falling ladder problem.

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    Lecture 42 - Mean Value Theorem

    Intuition behind the Mean Value Theorem.

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    Lecture 43 - Indefinite Integrals Part 1

    An introduction to indefinite integration of polynomials.

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    Lecture 44 - Indefinite Integrals Part 2

    Examples of taking the indefinite integral (or anti-derivative) of polynomials.

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    Lecture 45 - Indefinite Integrals Part 3

    Integration by doing the chain rule in reverse.

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    Lecture 46 - Indefinite Integrals Part 4

    Integration by substitution (or the reverse-chain-rule).

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    Lecture 47 - Indefinite Integrals Part 5

    Introduction to Integration by Parts (kind of the reverse-product rule).

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    Lecture 48 - Indefinite Integrals Part 6

    Example using Integration by Parts.

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    Lecture 49 - Indefinite Integrals Part 7

    Another example using integration by parts.

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    Lecture 50 - Another U-substitution Example

    Finding the antiderivative using u-substitution.

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    Lecture 51 - Definite Integrals Part 1

    Using the definite integral to solve for the area under a curve. Intuition on why the antiderivative is the same thing as the area under a curve.

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    Lecture 52 - Definite Integrals Part 2

    More on why the antiderivative and the area under a curve are essentially the same thing.

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    Lecture 53 - Definite Integrals Part 3

    More on why the antiderivative and the area under a curve are essentially the same thing.

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    Lecture 54 - Definite Integrals Part 4

    Examples of using definite integrals to find the area under a curve.

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    Lecture 55 - Definite Integrals Part 5

    More examples of using definite integrals to calculate the area between curves.

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    Lecture 56 - Definite Integral with Substitution

    Solving a definite integral with substitution (or the reverse chain rule).

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    Lecture 57 - Integrals: Trig Substitution Part 1

    Example of using trig substitution to solve an indefinite integral.

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    Lecture 58 - Integrals: Trig Substitution Part 2

    Another example of finding an anti-derivative using trigonometric substitution.

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    Lecture 59 - Integrals: Trig Substitution Part 3

    Example using trig substitution (and trig identities) to solve an integral.

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    Lecture 60 - Introduction to Differential Equations

    3 basic differential equations that can be solved by taking the antiderivatives of both sides.

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    Lecture 61 - Solid of Revolution Part 1

    Figuring out the volume of a function rotated about the x-axis.

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    Lecture 62 - Solid of Revolution Part 2

    The volume of y=sqrt(x) between x=0 and x=1 rotated around x-axis.

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    Lecture 63 - Solid of Revolution Part 3

    Figuring out the equation for the volume of a sphere.

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    Lecture 64 - Solid of Revolution Part 4

    More volumes around the x-axis.

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    Lecture 65 - Solid of Revolution Part 5

    Use the "shell method" to rotate about the y-axis.

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    Lecture 66 - Solid of Revolution Part 6

    Using the disk method around the y-axis.

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    Lecture 67 - Solid of Revolution Part 7

    Taking the revolution around something other than one of the axes.

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    Lecture 68 - Solid of Revolution Part 8

    The last part of the problem in part 7.

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    Lecture 69 - Polynomial Approximation of Functions Part 1

    Using a polynomial to approximate a function at f(0).

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    Lecture 70 - Polynomial Approximation of Functions Part 2

    Approximating a function with a polynomial by making the derivatives equal at f(0) (Maclauren Series).

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    Lecture 71 - Polynomial Approximation of Functions Part 3

    A glimpse of the mystery of the Universe as we approximate e^x with an infinite series.

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    Lecture 72 - Polynomial Approximation of Functions Part 4

    Approximating cos x with a Maclaurin series.

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    Lecture 73 - Polynomial Approximation of Functions Part 5

    MacLaurin representation of sin x.

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    Lecture 74 - Polynomial Approximation of Functions Part 6

    A pattern emerges!

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    Lecture 75 - Polynomial Approximation of Functions Part 7

    The most amazing conclusion in mathematics!

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    Lecture 76 - Taylor Polynomials

    Approximating a function with a Taylor Polynomial.

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    Lecture 77 - AP Calculus BC Exams: 2008 1 A

    Part 1a of the 2008 BC free response.

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    Lecture 78 - AP Calculus BC Exams: 2008 1 B & C

    Parts b and c of problem 1 (free response).

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    Lecture 79 - AP Calculus BC Exams: 2008 1 C & D

    Parts c&d of problem 1 in the 2008 AP Calculus BC free response.

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    Lecture 80 - AP Calculus BC Exams: 2008 1 D

    Part 1d of the 2008 AP Calculus BC exam (free response).

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    Lecture 81 - Calculus BC 2008 2 A

    2a of 2008 Calculus BC exam (free response).

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    Lecture 82 - Calculus BC 2008 2 B & C

    Parts 2b and 2c of the 2008 BC exam (free response).

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    Lecture 83 - Calculus BC 2008 2 D

    Part 2d of the 2008 Calculus BC exam free-response section.

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    Lecture 84 - Partial Derivatives Part 1

    Introduction to partial derivatives.

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    Lecture 85 - Partial Derivatives Part 2

    More on partial derivatives.

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    Lecture 86 - Gradient

    Introduction to the gradient.

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    Lecture 87 - Gradient of a Scalar Field

    Intuition of the gradient of a scalar field (temperature in a room) in 3 dimensions.

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    Lecture 88 - Divergence Part 1

    Introduction to the divergence of a vector field.

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    Lecture 89 - Divergence Part 2

    The intuition of what the divergence of a vector field is.

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    Lecture 90 - Divergence Part 3

    Analyzing a vector field using its divergence.

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    Lecture 91 - Curl Part 1

    Introduction to the curl of a vector field.

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    Lecture 92 - Curl Part 2

    The mechanics of calculating curl.

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    Lecture 93 - Curl Part 3

    More on curl.

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    Lecture 94 - Double Integrals Part 1

    Introduction to the double integral.

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    Lecture 95 - Double Integrals Part 2

    Figuring out the volume under z=xy^2.

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    Lecture 96 - Double Integrals Part 3

    Let's integrate dy first!

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    Lecture 97 - Double Integrals Part 4

    Another way to conceptualize the double integral.

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    Lecture 98 - Double Integrals Part 5

    Finding the volume when we have variable boundaries.

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    Lecture 99 - Double Integrals Part 6

    Let's evaluate the double integrals with y=x^2 as one of the boundaries.

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     Lecture 100 - Triple Integrals Part 1

     Introduction to the triple integral.

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     Lecture 101 - Triple Integrals Part 2

     Using a triple integral to find the mass of a volume of variable density.

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     Lecture 102 - Triple Integrals Part 3

     Figuring out the boundaries of integration.

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     Lecture 103 - (2^ln x)/x Antiderivative Example

     Finding ?(2^ln x)/x dx.

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     Lecture 104 - Line Integrals

     Introduction to the Line Integral.

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     Lecture 105 - Line Integral Example 1

     Concrete example using a line integral.

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     Lecture 106 - Line Integral Example 2 Part 1

     Line integral over a closed path (part 1).

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     Lecture 107 - Line Integral Example 2 Part 2

     Part 2 of an example of taking a line integral over a closed path.

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     Lecture 108 - Position Vector Valued Functions

     Using a position vector valued function to describe a curve or path.

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     Lecture 109 - Derivative of a Position Vector Valued Function

     Visualizing the derivative of a position vector valued function.

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     Lecture 110 - Differential of a Vector Valued Function

     Understanding the differential of a vector valued function.

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     Lecture 111 - Differential of a Vector Valued Function Example

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

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     Lecture 112 - Line Integrals and Vector Fields

     Using line integrals to find the work done on a particle moving through a vector field.

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     Lecture 113 - Using a Line Integral to Find a Vector Field Example

     Using a line integral to find the work done by a vector field example.

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     Lecture 114 - Parametrization of a Reverse Path

     Understanding how to parametrize a reverse path for the same curve.

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     Lecture 115 - Scalar Field Line Integral Independent of Path Direction

     Showing that the line integral of a scalar field is independent of path direction.

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     Lecture 116 - Vector Field Line Integral Dependent of Path Direction

     Showing that, unlike line integrals of scalar fields, line integrals over vector fields are path direction dependent.

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     Lecture 117 - Path Independence for Line Integrals

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

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     Lecture 118 - Closed Curve Line Integrals of Conservative Vector Fields

     Showing that the line integral along closed curves of conservative vector fields is zero.

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     Lecture 119 - Example of Closed Line Integral of Conservative Field

     Example of taking a closed line integral of a conservative field.

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     Lecture 120 - Second Example of Line Integral of Conservative Vector Field

     Using path independence of a conservative vector field to solve a line integral.

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     Lecture 121 - Green's Theorem Proof Part 1

     Part 1 of the proof of Green's Theorem.

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     Lecture 122 - Green's Theorem Proof Part 2

     Part 2 of the proof of Green's Theorem.

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     Lecture 123 - Green's Theorem Example Part 1

     Using Green's Theorem to solve a line integral of a vector field.

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     Lecture 124 - Green's Theorem Example Part 2

     Another example applying Green's Theorem.

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     Lecture 125 - Introduction to Parametrizing a Surface with Two Parameters

     Introduction to Parametrizing a Surface with Two Parameters.

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     Lecture 126 - Position Vector-Valued Function for a Parametrization of Two Parameters

     Determining a Position Vector-Valued Function for a Parametrization of Two Parameters.

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     Lecture 127 - Partial Derivatives of Vector-Valued Functions

     Partial Derivatives of Vector-Valued Functions.

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     Lecture 128 - Introduction to the Surface Integral

     Introduction to the surface integral.

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     Lecture 129 - Calculating a Surface Integral Example Part 1

     Example of calculating a surface integral part 1.

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     Lecture 130 - Calculating a Surface Integral Example Part 1

     Example of calculating a surface integral part 2.

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     Lecture 131 - Calculating a Surface Integral Example Part 1

     Example of calculating a surface integral part 3.

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     Lecture 132 - L'Hopital's Rule

     Introduction to L'Hopital's Rule.

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     Lecture 133 - L'Hopital's Rule Example 1

     L'Hopital's Rule Example 1.

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     Lecture 134 - L'Hopital's Rule Example 2

     L'Hopital's Rule Example 2.

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     Lecture 135 - L'Hopital's Rule Example 3

     L'Hopital's Rule Example 1.