Engineering Mathematics

The University of New South Wales , Prof.Chris Tisdell


Critical points + 2nd derivative test: Multivariable calculus


Lecture Description

I discuss and solve an example where the location and nature of critical points of a function of two variables is sought. The method is to calculate the partial derivatives, set them to zero and then solve to find the critical points. The critical points are then classified by employing the 2nd derivative test for functions of two variables.Such an example is seen in 2nd year university mathematics subjects.

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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