Lecture Details :
An introduction to the calculus of vector functions of one variable. Some simple problems are discussed, including differentiation, integration and how to determine the curve associated with a function (in this example, a helix). Such functions can be used to describe curves and motion in space.
Course Description :
Vector Revision - Intro to curves and vector functions - Limits of vector functions - Calculus of vector functions - Calculus of vector functions tutorial - Vector functions of one variable tutorial - Vector functions tutorial - Intro to functions of two variables - Partial derivatives-2 variable functions: graphs + limits tutorial - Multivariable chain rule and differentiability - Chain rule: partial derivative of $arctan (y/x)$ w.r.t. $x$ - Chain rule: identity involving partial derivatives - Chain rule & partial derivatives - Partial derivatives and PDEs tutorial - Multivariable chain rule tutorial - Gradient and directional derivative - Gradient of a function - Tutorial on gradient and tangent plane - Directional derivative of $f(x,y)$ - Gradient & directional derivative tutorial - Tangent plane approximation and error estimation - Partial derivatives and error estimation - Multivariable Taylor Polynomials - Taylor polynomials: functions of two variables - Differentiation under integral signs: Leibniz rule - Leibniz' rule: Integration via differentiation under integral sign
Evaluating challenging integrals via differentiation: Leibniz rule - Critical points of functions. Chris Tisdell UNSW Sydney - Second derivative test: two variables. Chris Tisdell UNSW Sydney - How to find critical points of functions - Critical points + 2nd derivative test: Multivariable calculus - Critical points + 2nd derivative test: Multivariable calculus - How to find and classify critical points of functions - Lagrange multipliers - Lagrange multipliers: Extreme values of a function subject to a constraint - Lagrange multipliers example - Lagrange multiplier example: Minimizing a function subject to a constraint - 2nd derivative test, max / min and Lagrange multipliers tutorial - Lagrange multipliers: 2 constraints-Intro to vector fields - What is the divergence - Divergence + Vector fields - Divergence of a vector field: Vector Calculus - What is the curl? Chris Tisdell UNSW Sydney - Curl of a vector field (ex. no.1): Vector Calculus - Line integrals - Integration over curves - Path integral (scalar line integral) from vector calculus