# Multivariable Calculus Tutorials from Khan Academy

## Video Lectures

Displaying all 86 video lectures.

Lecture 1Play Video |
Multivariable functionsAn introduction to multivariable functions, and a welcome to the multivariable calculus content as a whole. |

Lecture 2Play Video |
Representing points in 3dLearn how to represent and think about points and vectors in three-dimensional space. |

Lecture 3Play Video |
Introduction to 3d graphsThree-dimensional graphs are a way to represent functions with a two-dimensional input and a one-dimensional output. |

Lecture 4Play Video |
Interpreting graphs with slices3d graphs can be a lot to take in, but it helps to imagine slicing them with planes parallel to the x-axis or y-axis and relate them with two-dimensional graphs. |

Lecture 5Play Video |
Contour plotsAn alternative method to representing multivariable functions with a two-dimensional input and a one-dimensional output, contour maps involve drawing purely in the input space. |

Lecture 6Play Video |
Parametric curvesWhen a function has a one-dimensional input, but a multidimensional output, you can think of it as drawing a curve in space. |

Lecture 7Play Video |
Parametric surfacesFunctions that have a two-dimensional input and a three-dimensional output can be thought of as drawing a surface in three-dimensional space. This is actually pretty cool. |

Lecture 8Play Video |
Vector fields, introductionVector fields let you visualize a function with a two-dimensional input and a two-dimensional output. You end up with, well, a field of vectors sitting at various points in two-dimensional space. |

Lecture 9Play Video |
Fluid flow and vector fieldsA neat way to interpret a vector field is to imagine that it represents some kind of fluid flow. |

Lecture 10Play Video |
3d vector fields, introductionVector fields can also be three-dimensional, though this can be a bit trickier to visualize. |

Lecture 11Play Video |
3d vector field exampleSee an example of how you can start to understand how the formula for a three-dimensional vector field relates to the way it looks. |

Lecture 12Play Video |
Transformations, part 1One fun way to think about functions is to imagine that they literally move the points from the input space over to the output space. See what this looks like with some one-dimensional examples. |

Lecture 13Play Video |
Transformations, part 2More transformations, but this time with a function that maps two dimensions to two dimensions. |

Lecture 14Play Video |
Transformations, part 3Learn how you can think about a parametric surface as a certain kind of transformation. |

Lecture 15Play Video |
Partial derivatives, introductionPartial derivatives tell you how a multivariable function changes as you tweak just one of the variables in its input. |

Lecture 16Play Video |
Partial derivatives and graphsOne of the best ways to think about partial derivatives is by slicing the graph of a multivariable function. |

Lecture 17Play Video |
Formal definition of partial derivativesPartial derivatives are formally defined using a limit, much like ordinary derivatives. |

Lecture 18Play Video |
Symmetry of second partial derivativesThere are many ways to take a "second partial derivative", but some of them secretly turn out to be the same thing. |

Lecture 19Play Video |
GradientThe gradient captures all the partial derivative information of a scalar-valued multivariable function. |

Lecture 20Play Video |
Gradient and graphsLearn how the gradient can be thought of as pointing in the "direction of steepest ascent". This is a rather important interpretation for the gradient. |

Lecture 21Play Video |
Directional derivativeDirectional derivatives tell you how a multivariable function changes as you move along some vector in its input space. |

Lecture 22Play Video |
Directional derivative, formal definitionLearn the limit definition of a directional derivative. This helps to clarify what it is really doing. |

Lecture 23Play Video |
Directional derivatives and slopeThe directional derivative can be used to compute the slope of a slice of a graph, but you must be careful to use a unit vector. |

Lecture 24Play Video |
Why the gradient is the direction of steepest ascentThe way we compute the gradient seems unrelated to its interpretation as the direction of steepest ascent. Here you can see how the two relate. |

Lecture 25Play Video |
Gradient and contour mapsGradient vectors always point perpendicular to contour lines. |

Lecture 26Play Video |
Position vector valued functions | Multivariable Calculus | Khan AcademyUsing a position vector valued function to describe a curve or path Watch the next lesson: https://www.khanacademy.org/math/multivariable-calculus/line... Missed the previous lesson? https://www.khanacademy.org/math/multivariable-calculus/line... 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. Subscribe to KhanAcademy’s Multivariable Calculus channel: https://www.youtube.com/channel/UCQQZDc22yCogOyx6DwXt-Ig?sub... |

Lecture 27Play Video |
Derivative of a position vector valued function | Multivariable Calculus | Khan AcademyVisualizing the derivative of a position vector valued function Watch the next lesson: https://www.khanacademy.org/math/multivariable-calculus/line... Missed the previous lesson? https://www.khanacademy.org/math/multivariable-calculus/line... 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. Subscribe to KhanAcademy’s Multivariable Calculus channel: https://www.youtube.com/channel/UCQQZDc22yCogOyx6DwXt-Ig?sub... |

Lecture 28Play Video |
Differential of a vector valued function | Multivariable Calculus | Khan AcademyUnderstanding the differential of a vector valued function Watch the next lesson: https://www.khanacademy.org/math/multivariable-calculus/line... Missed the previous lesson? https://www.khanacademy.org/math/multivariable-calculus/line... 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. Subscribe to KhanAcademy’s Multivariable Calculus channel: https://www.youtube.com/channel/UCQQZDc22yCogOyx6DwXt-Ig?sub... |

Lecture 29Play Video |
Vector valued function derivative example | Multivariable Calculus | Khan AcademyConcrete example of the derivative of a vector valued function to better understand what it means Watch the next lesson: https://www.khanacademy.org/math/multivariable-calculus/line... Missed the previous lesson? https://www.khanacademy.org/math/multivariable-calculus/line... 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. Subscribe to KhanAcademy’s Multivariable Calculus channel: https://www.youtube.com/channel/UCQQZDc22yCogOyx6DwXt-Ig?sub... |

Lecture 30Play Video |
Multivariable chain ruleThis is the simplest case of taking the derivative of a composition involving multivariable functions. |

Lecture 31Play Video |
Multivariable chain rule intuitionGet a feel for what the multivariable is really saying, and how thinking about various "nudges" in space makes it intuitive. |

Lecture 32Play Video |
Vector form of the multivariable chain ruleThe multivariable chain rule is more often expressed in terms of the gradient and a vector-valued derivative. This makes it look very analogous to the single-variable chain rule. |

Lecture 33Play Video |
Multivariable chain rule and directional derivativesSee how the multivariable chain rule can be expressed in terms of the directional derivative. |

Lecture 34Play Video |
More formal treatment of multivariable chain ruleFor those of you who want to see how the multivariable chain rule looks in the context of the limit definitions of various forms of the derivative. |

Lecture 35Play Video |
Curvature intuitionAn introduction to curvature, the radius of curvature, and how you can think about each one geometrically. |

Lecture 36Play Video |
Curvature formula, part 1Curvature is computed by first finding a unit tangent vector function, then finding its derivative with respect to arc length. Here we start thinking about what that means. |

Lecture 37Play Video |
Curvature formula, part 2A continuation in explaining how curvature is computed, with the formula for a circle as a guiding example. |

Lecture 38Play Video |
Curvature formula, part 3Here, this concludes the explanation for how curvature is the derivative of a unit tangent vector with respect to length. |

Lecture 39Play Video |
Curvature formula, part 4After the last video made reference to an explicit curvature formula, here you can start to get an intuition for why that seemingly unrelated formula describes curvature. |

Lecture 40Play Video |
Curvature formula, part 5Here, we finish the intuition for how the curvature relates to the cross product between the first two derivatives of a parametric function. |

Lecture 41Play Video |
Curvature of a helix, part 1An example of computing curvature by finding the unit tangent vector function, then computing its derivative with respect to arc length. |

Lecture 42Play Video |
Curvature of a helix, part 2This finishes up the helix-curvature example started in the last video. |

Lecture 43Play Video |
Curvature of a cycloidAn example of computing curvature with the explicit formula. |

Lecture 44Play Video |
Computing the partial derivative of a vector-valued functionWhen a function has a multidimensional input, and a multidimensional output, you can take its partial derivative by computing the partial derivative of each component in the output. |

Lecture 45Play Video |
Partial derivative of a parametric surface, part 1When a vector-valued function represents a parametric surface, how do you interpret its partial derivative? |

Lecture 46Play Video |
Partial derivative of a parametric surface, part 2Taking the same example surface used in the last example, we now take a look at the partial derivative in the other direction. |

Lecture 47Play Video |
Partial derivatives of vector fieldsHow do you intepret the partial derivatives of the function which defines a vector field? |

Lecture 48Play Video |
Partial derivatives of vector fields, component by componentHere we step through each partial derivative of each component in a vector field, and understand what each means geometrically. |

Lecture 49Play Video |
Divergence intuition, part 1Vector fields can be thought of as representing fluid flow, and divergence is all about studying the change in fluid density during that flow. |

Lecture 50Play Video |
Divergence intuition, part 2In preparation for finding the formula for divergence, we start getting an intuition for what points of positive, negative and zero divergence should look like. |

Lecture 51Play Video |
Divergence formula, part 1How does the x-component of a vector field relate to the divergence? |

Lecture 52Play Video |
Divergence formula, part 2Here we finish the line of reasoning which leads to the formula for divergence in two dimensions. |

Lecture 53Play Video |
Divergence exampleAn example of computing and interpreting the divergence of a two-dimensional vector field. |

Lecture 54Play Video |
Divergence notationLearn how divergence is expressed using the same upsidedown triangle symbols that the gradient uses. |

Lecture 55Play Video |
2d curl intuitionA description of how vector fields relate to fluid rotation, laying the intuition for what the operation of curl represents. |

Lecture 56Play Video |
2d curl formulaHere we build up to the formula for computing the two-dimensional curl of a vector field, reasoning through what partial derivative information corresponds to fluid rotation. |

Lecture 57Play Video |
2d curl exampleA worked example of computing and interpreting two-dimensional curl. |

Lecture 58Play Video |
2d curl nuanceThe meaning of positive curl in a fluid flow can sometimes look a bit different from the clear cut rotation-around-a-point examples discussed in previous videos. |

Lecture 59Play Video |
Describing rotation in 3d with a vectorLearn how a three-dimensional vector can be used to describe three-dimensional rotation. This is important for understanding three-dimensional curl. |

Lecture 60Play Video |
3d curl intuition, part 1Here we start transitioning from the understanding of two-dimensional curl into an understanding of three-dimensional curl. |

Lecture 61Play Video |
3d curl intuition, part 2Continuing the intuition for how three-dimensional curl represents rotation in three-dimensional fluid flow. |

Lecture 62Play Video |
3d curl formula, part 1How to compute a three-dimensional curl, imagined as a cross product of sorts. |

Lecture 63Play Video |
3d curl formula, part 2This finishes the demonstration of how to compute three-dimensional curl using a certain determinant. |

Lecture 64Play Video |
3d curl computation exampleA worked example of a three-dimensional curl computation. |

Lecture 65Play Video |
Laplacian intuitionA visual understanding for how the Laplace operator is an extension of the second derivative to multivariable functions. |

Lecture 66Play Video |
Laplacian computation exampleA worked example of computing the laplacian of a two-variable function. |

Lecture 67Play Video |
Explicit Laplacian formulaThis is another way you might see the Laplace operator written. |

Lecture 68Play Video |
Harmonic FunctionsIf the Laplacian of a function is zero everywhere, it is called Harmonic. Harmonic functions arise all the time in physics, capturing a certain notion of "stability", whenever one point in space is influenced by its neighbors. |

Lecture 69Play Video |
What is a tangent planeThe "tangent plane" of the graph of a function is, well, a two-dimensional plane that is tangent to this graph. Here you can see what that looks like. |

Lecture 70Play Video |
Controlling a plane in spaceHow can you describe a specified plane in space as the graph of a function? |

Lecture 71Play Video |
Computing a tangent planeHere you can see how to use the control over functions whose graphs are planes, as introduced in the last video, to find the tangent plane to a function graph. |

Lecture 72Play Video |
Local linearizationA "local linearization" is the generalization of tangent plane functions; one that can apply to multivariable functions with any number of inputs. |

Lecture 73Play Video |
What do quadratic approximations look likeAfter learning about local linearizations of multivariable functions, the next step is to understand how to approximate a function even more closely with a quadratic approximation. |

Lecture 74Play Video |
Quadratic approximation formula, part 1How to creat a quadratic function that approximates an arbitrary two-variable function. |

Lecture 75Play Video |
Quadratic approximation formula, part 2A continuation from the previous video, leading to the full formula for the quadratic approximation of a two-variable function. |

Lecture 76Play Video |
Quadratic approximation exampleA worked example for finding the quadratic approximation of a two-variable function. |

Lecture 77Play Video |
The Hessian matrixThe Hessian matrix is a way of organizing all the second partial derivative information of a multivariable function. |

Lecture 78Play Video |
Expressing a quadratic form with a matrixHow to write an expression like ax^2 + bxy + cy^2 using matrices and vectors. |

Lecture 79Play Video |
Vector form of multivariable quadratic approximationThis is the more general form of a quadratic approximation for a scalar-valued multivariable function. It is analogous to a quadratic Taylor polynomial in the single-variable world. |

Lecture 80Play Video |
Multivariable maxima and minimaA description of maxima and minima of multivariable functions, what they look like, and a little bit about how to find them. |

Lecture 81Play Video |
Saddle pointsJust because the tangent plane to a multivariable function is flat, it doesn't mean that point is a local minimum or a local maximum. There is a third possibility, new to multivariable calculus, called a "saddle point". |

Lecture 82Play Video |
Warm up to the second partial derivative testAn example of looking for local minima in a multivariable function by finding where tangent planes are flat, along with some of the intuitions that will underly the second partial derivative test. |

Lecture 83Play Video |
Second partial derivative testHow to determine if the critical point of a two-variable function is a local minimum, a local maximum, or a saddle point. |

Lecture 84Play Video |
Second partial derivative test intuitionThe second partial derivative test is based on a formula which seems to come out of nowhere. Here, you can see a little more intuition for why it looks the way it does. |

Lecture 85Play Video |
Second partial derivative test example, part 1A worked example of finding a classifying critical points of a two-variable function. |

Lecture 86Play Video |
Second partial derivative test example, part 2Continuing the worked example from the previous video, now classifying each critical point. |