Part 2 of an example of taking a line integral over a closed path Watch the next lesson: https://www.khanacademy.org/math/multivariable-calculus/line_integrals_topic/position_vector_functions/v/position-vector-valued-functions?utm_source=YT&utm_medium=Desc&utm_campaign=MultivariableCalculus Missed the previous lesson? https://www.khanacademy.org/math/multivariable-calculus/line_integrals_topic/line_integrals/v/line-integral-example-2-part-1?utm_source=YT&utm_medium=Desc&utm_campaign=MultivariableCalculus 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. About Khan Academy: Khan Academy offers practice exercises, instructional videos, and a personalized learning dashboard that empower learners to study at their own pace in and outside of the classroom. We tackle math, science, computer programming, history, art history, economics, and more. Our math missions guide learners from kindergarten to calculus using state-of-the-art, adaptive technology that identifies strengths and learning gaps. We've also partnered with institutions like NASA, The Museum of Modern Art, The California Academy of Sciences, and MIT to offer specialized content. ... https://www.youtube.com/watch?v=Qqanbd3gLhw Note: All Khan Academy content is available for free at (www.khanacademy.org)

Showing that matrix products are associative Watch the next lesson: https://www.khanacademy.org/math/linear-algebra/matrix_transformations/composition_of_transformations/v/distributive-property-of-matrix-products?utm_source=YT&utm_medium=Desc&utm_campaign=LinearAlgebra Missed the previous lesson? https://www.khanacademy.org/math/linear-algebra/matrix_transformations/composition_of_transformations/v/linear-algebra-matrix-product-examples?utm_source=YT&utm_medium=Desc&utm_campaign=LinearAlgebra Linear Algebra on Khan Academy: Have you ever wondered what the difference is between speed and velocity? Ever try to visualize in four dimensions or six or seven? Linear algebra describes things in two dimensions, but many of the concepts can be extended into three, four or more. Linear algebra implies two dimensional reasoning, however, the concepts covered in linear algebra provide the basis for multi-dimensional representations of mathematical reasoning. Matrices, vectors, vector spaces, transformations, eigenvectors/values all help us to visualize and understand multi dimensional concepts. This is an advanced course normally taken by science or engineering majors after taking at least two semesters of calculus (although calculus really isn't a prereq) so don't confuse this with regular high school algebra. About Khan Academy: Khan Academy offers practice exercises, instructional videos, and a personalized learning dashboard that empower learners to study at their own pace in and outside of the classroom. We tackle math, science, computer programming, history, art history, economics, and more. Our math missions guide learners from kindergarten to calculus using state-of-the-art, adaptive technology that identifies strengths and learning gaps. We've also partnered with institutions like NASA, The Museum of Modern Art, The California Academy of Sciences, and MIT to offer specialized content. ... https://www.youtube.com/watch?v=Hhc96U_HvQE Note: All Khan Academy content is available for free at (www.khanacademy.org)

Enjoy the videos and music you love, upload original content, and share it all with friends, family, and the world on YouTube. ... https://www.youtube.com/watch?v=pUzHSQa6lko Note: All Khan Academy content is available for free at (www.khanacademy.org)

More free lessons at: http://www.khanacademy.org/video?v=ssz6TElXEOM ... https://www.youtube.com/watch?v=ssz6TElXEOM Note: All Khan Academy content is available for free at (www.khanacademy.org)