# Engineering Mathematics

The University of New South Wales Course , Prof. Chris Tisdell

### Lecture 15: Partial derivatives and PDEs tutorial

##### Lecture Details :

This is basic tutorial on how to calculate partial derivatives. The ideas are applied to show that certain functions satisfy a famous partial differential equation, known as the wave equation. Such ideas are seen in university mathematics.

##### Course Description :

Contents:
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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