Engineering Mathematics

The University of New South Wales , Prof.Chris Tisdell


Calculus of vector functions - 1 variable. Chris Tisdell UNSW Sydney


Lecture Description

This lecture introduces the idea of derivative and integral of vector-valued functions of one variable. We discuss the geometric significance and show how to compute the derivative and integral via examples. We briefly discuss an application to calculating arc length.

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

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