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source: UNSWelearning 2009年5月10日
In this lecture I discuss the applications of multiple integrals in an applied mathematics and engineering context. I discuss how to calculate the mass, moments and centre of mass of 2-dimensional thin plates. I also briefly glimpse at applications of triple integrals.
This is a series of lectures for "Several Variable Calculus" and "Vector Calculus", which is a 2nd-year mathematics subject taught at UNSW, Sydney. This playlist provides a shapshot of some lectures presented in Session 1, 2009 and Session 1, 2011..
These lectures focus on presenting vector calculus in an applied and engineering context, while maintaining mathematical rigour. Thus, this playlist may be useful to students of mathematics, but also to those of engineering, physics and the applied sciences. There is an emphasis on examples and also on proofs!
Applications of Double integrals. 44:29
Path integrals - How to integrate over curves 46:51
What is a vector field?? 42:47
What is the divergence? 46:22
What is the Curl? 48:35
What is a line integral? 48:33
Applications of Line Integrals. 46:20
Fundamental theorem of line integrals. 40:28
What is Green's theorem? 47:50
Green's Theorem 39:14
Parametrised surfaces. 32:49
What is a surface integral? (part 1) 41:39
More on surface integrals. 30:48
Surface integrals + vector fields. 25:01
Divergence theorem of Gauss 12:21
How to solve PDEs via separation of variables + Fourier series. 42:11
Vector Revision 42:51
Intro to curves and vector functions 49:07
Limits of vector functions 44:38
Calculus of vector functions - 1 variable 20:42
Calculus of vector functions tutorial 44:25
Vector functions tutorial 29:07
Intro to functions of two variables 33:59
Limits of functions of two variables 48:53
Partial derivatives 45:48
Partial derivatives and PDEs tutorial 9:23
2 variable functions: graphs + limits tutorial 41:26
Multivariable chain rule and differentiability 48:54
Chain rule: partial derivative of $\arctan (y/x)$ w.r.t. $x$ 5:33
Chain rule & partial derivatives 9:01
Chain rule: identity involving partial derivatives 7:43
Multivariable chain rule tutorial 33:53
Leibniz' rule: Integration via differentiation under integral sign 5:39
Evaluating challenging integrals via differentiation: Leibniz rule 8:03
Gradient and directional derivative 1:07:26
Gradient & directional derivative tutorial 45:41
Gradient & directional derivative tutorial 45:41
Directional derivative of f(x,y) 6:49
Tangent plane approximation and error estimation 28:19
Tutorial on gradient and tangent plane 22:59
Partial derivatives and error estimation 12:21
Multivariable Taylor Polynomials 54:33
Taylor polynomials: functions of two variables 10:44
Limits, chain rule, arc length. Multivariable calculus. 34:54
Critical points of functions 30:42
How to find critical points of functions 14:40
How to find critical points of functions 14:40
Second derivative test: two variables 27:10
Critical points + 2nd derivative test: Multivariable calculus 7:17
How to find and classify critical points of functions 11:58
Lagrange multipliers 45:23
Lagrange multipliers: 2 constraints 14:24
Lagrange multipliers: Extreme values of a function subject to a constraint 7:31
Lagrange multipliers example 11:08
Lagrange multiplier example: Minimizing a function subject to a constraint 8:29
2nd derivative test, max / min and Lagrange multipliers tutorial 42:38
Intro to Jacobian + differentiability 41:53
Jacobian chain rule and inverse function theorem 26:59
Intro to double integrals 29:21
Double integrals over general regions 41:00
Double integrals: Volume between two surfaces 7:14
Double integrals: Volume of a tetrahedron 5:02
Double integral tutorial 11:12
Double integrals and area 10:13
Double integrals in polar co-ordinates 16:21
Reversing order in double integrals 11:55
Double integrals: reversing the order of integration 8:28
Applications of double integrals. 44:43
Double integrals and polar co-ordinates 35:48
Tutorial on double integrals 26:50
Centroid + double integral tutorial 25:03
Center of mass, double integrals and polar co-ordinates tutorial 33:34
Triple integral tutorial 39:34
Triple integrals in Cylindrical and Spherical Coordinates 40:01
Triple integrals & Center of Mass 27:53
Change of variables in double integrals tutorial 33:38
Path integral (scalar line integral) from vector calculus 6:15
Line integral example in 3D-space 5:55
Line integral from vector calculus over a closed curve 8:01
Line integral example from Vector Calculus 7:29
Divergence of a vector field: Vector Calculus 6:21
Curl of a vector field (ex. no.1): Vector Calculus 5:22
Curl of a vector field (ex. no.2): Vector calculus 8:38
Divergence theorem of Gauss 12:21
Intro to Fourier series and how to calculate them 13:53
How to compute a Fourier series: an example 8:25
What are Fourier series? 45:11
Tutorial on Fourier series 33:08
Fourier series + differential equations 17:48
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