  # Multivariable Calculus Tutorials from Khan Academy

## Video Lectures

Displaying all 86 video lectures.
 Lecture 1 Play Video Multivariable functionsAn introduction to multivariable functions, and a welcome to the multivariable calculus content as a whole. Lecture 2 Play Video Representing points in 3dLearn how to represent and think about points and vectors in three-dimensional space. Lecture 3 Play Video Introduction to 3d graphsThree-dimensional graphs are a way to represent functions with a two-dimensional input and a one-dimensional output. Lecture 4 Play 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 5 Play 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 6 Play 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 7 Play 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 8 Play 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 9 Play 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 10 Play Video 3d vector fields, introductionVector fields can also be three-dimensional, though this can be a bit trickier to visualize. Lecture 11 Play 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 12 Play 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 13 Play Video Transformations, part 2More transformations, but this time with a function that maps two dimensions to two dimensions. Lecture 14 Play Video Transformations, part 3Learn how you can think about a parametric surface as a certain kind of transformation. Lecture 15 Play Video Partial derivatives, introductionPartial derivatives tell you how a multivariable function changes as you tweak just one of the variables in its input. Lecture 16 Play Video Partial derivatives and graphsOne of the best ways to think about partial derivatives is by slicing the graph of a multivariable function. Lecture 17 Play Video Formal definition of partial derivativesPartial derivatives are formally defined using a limit, much like ordinary derivatives. Lecture 18 Play 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 19 Play Video GradientThe gradient captures all the partial derivative information of a scalar-valued multivariable function. Lecture 20 Play 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 21 Play Video Directional derivativeDirectional derivatives tell you how a multivariable function changes as you move along some vector in its input space. Lecture 22 Play Video Directional derivative, formal definitionLearn the limit definition of a directional derivative. This helps to clarify what it is really doing. Lecture 23 Play 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 24 Play 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 25 Play Video Gradient and contour mapsGradient vectors always point perpendicular to contour lines. Lecture 26 Play 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 27 Play 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 28 Play 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 29 Play 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 30 Play Video Multivariable chain ruleThis is the simplest case of taking the derivative of a composition involving multivariable functions. Lecture 31 Play 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 32 Play 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 33 Play Video Multivariable chain rule and directional derivativesSee how the multivariable chain rule can be expressed in terms of the directional derivative. Lecture 34 Play 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 35 Play Video Curvature intuitionAn introduction to curvature, the radius of curvature, and how you can think about each one geometrically. Lecture 36 Play 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 37 Play Video Curvature formula, part 2A continuation in explaining how curvature is computed, with the formula for a circle as a guiding example. Lecture 38 Play 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 39 Play 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 40 Play 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 41 Play 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 42 Play Video Curvature of a helix, part 2This finishes up the helix-curvature example started in the last video. Lecture 43 Play Video Curvature of a cycloidAn example of computing curvature with the explicit formula. Lecture 44 Play 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 45 Play 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 46 Play 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 47 Play Video Partial derivatives of vector fieldsHow do you intepret the partial derivatives of the function which defines a vector field? Lecture 48 Play 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 49 Play 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 50 Play 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 51 Play Video Divergence formula, part 1How does the x-component of a vector field relate to the divergence? Lecture 52 Play Video Divergence formula, part 2Here we finish the line of reasoning which leads to the formula for divergence in two dimensions. Lecture 53 Play Video Divergence exampleAn example of computing and interpreting the divergence of a two-dimensional vector field. Lecture 54 Play Video Divergence notationLearn how divergence is expressed using the same upsidedown triangle symbols that the gradient uses. Lecture 55 Play Video 2d curl intuitionA description of how vector fields relate to fluid rotation, laying the intuition for what the operation of curl represents. Lecture 56 Play 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 57 Play Video 2d curl exampleA worked example of computing and interpreting two-dimensional curl. Lecture 58 Play 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 59 Play 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 60 Play Video 3d curl intuition, part 1Here we start transitioning from the understanding of two-dimensional curl into an understanding of three-dimensional curl. Lecture 61 Play Video 3d curl intuition, part 2Continuing the intuition for how three-dimensional curl represents rotation in three-dimensional fluid flow. Lecture 62 Play Video 3d curl formula, part 1How to compute a three-dimensional curl, imagined as a cross product of sorts. Lecture 63 Play Video 3d curl formula, part 2This finishes the demonstration of how to compute three-dimensional curl using a certain determinant. Lecture 64 Play Video 3d curl computation exampleA worked example of a three-dimensional curl computation. Lecture 65 Play Video Laplacian intuitionA visual understanding for how the Laplace operator is an extension of the second derivative to multivariable functions. Lecture 66 Play Video Laplacian computation exampleA worked example of computing the laplacian of a two-variable function. Lecture 67 Play Video Explicit Laplacian formulaThis is another way you might see the Laplace operator written. Lecture 68 Play 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 69 Play 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 70 Play Video Controlling a plane in spaceHow can you describe a specified plane in space as the graph of a function? Lecture 71 Play 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 72 Play 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 73 Play 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 74 Play Video Quadratic approximation formula, part 1How to creat a quadratic function that approximates an arbitrary two-variable function. Lecture 75 Play 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 76 Play Video Quadratic approximation exampleA worked example for finding the quadratic approximation of a two-variable function. Lecture 77 Play Video The Hessian matrixThe Hessian matrix is a way of organizing all the second partial derivative information of a multivariable function. Lecture 78 Play Video Expressing a quadratic form with a matrixHow to write an expression like ax^2 + bxy + cy^2 using matrices and vectors. Lecture 79 Play 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 80 Play 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 81 Play 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 82 Play 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 83 Play 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 84 Play 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 85 Play Video Second partial derivative test example, part 1A worked example of finding a classifying critical points of a two-variable function. Lecture 86 Play Video Second partial derivative test example, part 2Continuing the worked example from the previous video, now classifying each critical point.