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Course

Subject:	Mathematics
Topic:	Multivariable Calculus
Views:	97,356

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Name:	Khan Academy (Khan)
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Multivariable Calculus Tutorials from Khan Academy

Video Lectures

Displaying all 86 video lectures.

Lecture 1 Play Video	Multivariable functions An introduction to multivariable functions, and a welcome to the multivariable calculus content as a whole.
Lecture 2 Play Video	Representing points in 3d Learn how to represent and think about points and vectors in three-dimensional space.
Lecture 3 Play Video	Introduction to 3d graphs Three-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 slices 3d 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 plots An 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 curves When 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 surfaces Functions 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, introduction Vector 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 fields A 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, introduction Vector fields can also be three-dimensional, though this can be a bit trickier to visualize.
Lecture 11 Play Video	3d vector field example See 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 1 One 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 2 More transformations, but this time with a function that maps two dimensions to two dimensions.
Lecture 14 Play Video	Transformations, part 3 Learn how you can think about a parametric surface as a certain kind of transformation.
Lecture 15 Play Video	Partial derivatives, introduction Partial 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 graphs One 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 derivatives Partial derivatives are formally defined using a limit, much like ordinary derivatives.
Lecture 18 Play Video	Symmetry of second partial derivatives There 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	Gradient The gradient captures all the partial derivative information of a scalar-valued multivariable function.
Lecture 20 Play Video	Gradient and graphs Learn 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 derivative Directional 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 definition Learn the limit definition of a directional derivative. This helps to clarify what it is really doing.
Lecture 23 Play Video	Directional derivatives and slope The 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 ascent The 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 maps Gradient vectors always point perpendicular to contour lines.
Lecture 26 Play Video	Position vector valued functions \| Multivariable Calculus \| Khan Academy Using 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 Academy Visualizing 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 Academy Understanding 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 Academy Concrete 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 rule This is the simplest case of taking the derivative of a composition involving multivariable functions.
Lecture 31 Play Video	Multivariable chain rule intuition Get 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 rule The 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 derivatives See how the multivariable chain rule can be expressed in terms of the directional derivative.
Lecture 34 Play Video	More formal treatment of multivariable chain rule For 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 intuition An introduction to curvature, the radius of curvature, and how you can think about each one geometrically.
Lecture 36 Play Video	Curvature formula, part 1 Curvature 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 2 A continuation in explaining how curvature is computed, with the formula for a circle as a guiding example.
Lecture 38 Play Video	Curvature formula, part 3 Here, 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 4 After 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 5 Here, 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 1 An 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 2 This finishes up the helix-curvature example started in the last video.
Lecture 43 Play Video	Curvature of a cycloid An example of computing curvature with the explicit formula.
Lecture 44 Play Video	Computing the partial derivative of a vector-valued function When 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 1 When 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 2 Taking 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 fields How 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 component Here 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 1 Vector 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 2 In 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 1 How does the x-component of a vector field relate to the divergence?
Lecture 52 Play Video	Divergence formula, part 2 Here we finish the line of reasoning which leads to the formula for divergence in two dimensions.
Lecture 53 Play Video	Divergence example An example of computing and interpreting the divergence of a two-dimensional vector field.
Lecture 54 Play Video	Divergence notation Learn how divergence is expressed using the same upsidedown triangle symbols that the gradient uses.
Lecture 55 Play Video	2d curl intuition A 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 formula Here 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 example A worked example of computing and interpreting two-dimensional curl.
Lecture 58 Play Video	2d curl nuance The 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 vector Learn 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 1 Here 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 2 Continuing the intuition for how three-dimensional curl represents rotation in three-dimensional fluid flow.
Lecture 62 Play Video	3d curl formula, part 1 How to compute a three-dimensional curl, imagined as a cross product of sorts.
Lecture 63 Play Video	3d curl formula, part 2 This finishes the demonstration of how to compute three-dimensional curl using a certain determinant.
Lecture 64 Play Video	3d curl computation example A worked example of a three-dimensional curl computation.
Lecture 65 Play Video	Laplacian intuition A visual understanding for how the Laplace operator is an extension of the second derivative to multivariable functions.
Lecture 66 Play Video	Laplacian computation example A worked example of computing the laplacian of a two-variable function.
Lecture 67 Play Video	Explicit Laplacian formula This is another way you might see the Laplace operator written.
Lecture 68 Play Video	Harmonic Functions If 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 plane The "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 space How can you describe a specified plane in space as the graph of a function?
Lecture 71 Play Video	Computing a tangent plane Here 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 linearization A "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 like After 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 1 How to creat a quadratic function that approximates an arbitrary two-variable function.
Lecture 75 Play Video	Quadratic approximation formula, part 2 A 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 example A worked example for finding the quadratic approximation of a two-variable function.
Lecture 77 Play Video	The Hessian matrix The 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 matrix How to write an expression like ax^2 + bxy + cy^2 using matrices and vectors.
Lecture 79 Play Video	Vector form of multivariable quadratic approximation This 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 minima A 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 points Just 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 test An 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 test How 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 intuition The 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 1 A worked example of finding a classifying critical points of a two-variable function.
Lecture 86 Play Video	Second partial derivative test example, part 2 Continuing the worked example from the previous video, now classifying each critical point.