Lecture 4: Introduction to Line Integrals
Lecture 4: Introduction to Line Integrals
Video Description: There is more than one type of integral in multivariable calculus. In this lesson, Herb Gross defines and discusses line integrals. He reviews integration with respect to a curve (line) as distinguished from an integral as an area computation (double ...
Lecture 2: The Fundamental Theorem
Lecture 2: The Fundamental Theorem
Video Description: Herb Gross illustrates the equivalence of the Fundamental Theorem of the Calculus of one variable to the Fundamental Theorem of Calculus for several variables. Topics include: The anti-derivative and the value of a definite integral; Iterated integrals. Instructor/speaker: ...
Lecture 5: Maxima and Minima in Several Variables
Lecture 5: Maxima and Minima in Several Variables
Video Description: With our knowledge of matrix algebra to help, Herb Gross teaches how to find the local maxima and minima of functions of several real variables. Instructor/speaker: Prof. Herbert Gross
Lecture 3: Directional Derivatives
Lecture 3: Directional Derivatives
Video Description: Herb Gross defines the directional derivative and demonstrates how to calculate it, emphasizing the importance of this topic in the study of Calculus of Several Variables. He also covers the definition of a gradient vector. Instructor/speaker: Prof. Herbert Gross
Lecture 4: Inverting More General Systems of Equations
Lecture 4: Inverting More General Systems of Equations
Video Description: Herb Gross demonstrates how to invert systems of non-linear equations. Instructor/speaker: Prof. Herbert Gross
Lecture 6: Exact Differentials
Lecture 6: Exact Differentials
Video Description: Herb Gross explains the necessary and sufficient condition for an expression of the form Mdx + Ndy to be an exact differential. Instructor/speaker: Prof. Herbert Gross
Lecture 4: Vectors in Polar Coordinates
Lecture 4: Vectors in Polar Coordinates
Video Description: Herb Gross describes vectors and polar coordinates. Topics include: Motion in the plane; Velocity and acceleration of polar coordinates. Instructor/speaker: Prof. Herbert Gross
Lecture 6: Equations of Lines & Planes
Lecture 6: Equations of Lines & Planes
Video Description: Herb Gross discusses the topic of equations of lines and planes in 3-dimensional space. Topics include: The normal vector to a plane; Parallel planes; Equation of a plane; Equation of a line in space. Instructor/speaker: Prof. Herbert Gross
Lecture 1: Linearity Revisited
Lecture 1: Linearity Revisited
Video Description: Herb Gross considers linearity in spaces of dimension greater than 2, including a discussion of local linearity. Instructor/speaker: Prof. Herbert Gross
Lecture 2: The "Game" of Matrices
Lecture 2: The "Game" of Matrices
Video Description: Herb Gross describes the "game" of matrices — the rules of matrix arithmetic and algebra. He also covers non-singularity and the inverse of a matrix. Instructor/speaker: Prof. Herbert Gross
Lecture 4: The Chain Rule
Lecture 4: The Chain Rule
Video Description: Herb Gross shows examples of the chain rule for several variables and develops a proof of the chain rule. He also explains how the chain rule works with higher order partial derivatives and mixed partial derivatives. Instructor/speaker: Prof. Herbert Gross
Lecture 5: Integrals Involving Parameters
Lecture 5: Integrals Involving Parameters
Video Description: Herb Gross show how the chain rule is involved in finding some integrals involving parameters. He computes the derivatives of integrals with constant limits, as well as derivatives of integrals with variable limits of integration (chain rule). ...
Lecture 2: "Arrow" Arithmetic
Lecture 2: "Arrow" Arithmetic
Video Description: Herb Gross reviews the definition of vectors — objects that have magnitude, direction, and sense. He also defines equality of vectors, their components, and rules of arithmetic. Vector arithmetic shares many structural properties with scalar arithmetic including a zero ...
Lecture 3: Three-Dimensional Vectors
Lecture 3: Three-Dimensional Vectors
Video Description: Herb Gross introduces 3-dimensional vectors — those with 3 othogonal components (x, y, z directions). This video also covers 3-dimensional magnitude and arithmetic in Cartesian coordinates (not polar coordinates). Prof. Gross also explains how 3-dimensional vectors are ...
Lecture 5: Green's Theorem
Lecture 5: Green's Theorem
Video Description: Herb Gross discusses Green's Theorem: The relationship between line integrals and double integrals over a region in 2-space. Instructor/speaker: Prof. Herbert Gross
Lecture 1: Double Multiple Sums
Lecture 1: Double Multiple Sums
Video Description: Herb Gross teaches us how to calculate infinite double (multiple) sums (for topics in calculus of several variables). This topic is analogous to the use of infinite sums in calculus of a single variable. Instructor/speaker: Prof. Herbert Gross
Lecture 3: Multiple Integration and the Jacobian
Lecture 3: Multiple Integration and the Jacobian
Video Description: Herb Gross illustrates how the Jacobian arises when changing coordinates in order to calculate a double integral. Instructor/speaker: Prof. Herbert Gross
Lecture 1: Vector Functions of a Scalar Variable
Lecture 1: Vector Functions of a Scalar Variable
Video Description: Herb Gross introduces the topic of vector functions of scalar variables and explains why the result is a vector (since the inputs are scalars). He then reviews limits (definition and related theorems) and presents the topic of differentiation of ...
Lecture 3: Inverting a Matrix
Lecture 3: Inverting a Matrix
Video Description: Herb Gross illustrates how to invert a square matrix and its relevance to Calculus of Several Variables. Instructor/speaker: Prof. Herbert Gross
Lecture 3: Polar Coordinates
Lecture 3: Polar Coordinates
Video Description: Herb Gross defines and demonstrates the use of polar coordinates. He describes the non-uniqueness of polar coordinates and how to calculate the slope of a curve, which depends on the angle the curve makes with the radius vector. Finally, he computes the area (in terms of ...