math150L2_part1

# math150L2_part1 - Section I - Introduction 1. Preliminaries...

This preview shows pages 1–4. Sign up to view the full content.

Section I - Introduction 1. Preliminaries 1.1 Function of single variable, y = f ( x ) . y is called dependent variable . x is called independent variable . Derivative of y deﬁned as: dy dx = lim 4 x 0 f ( x + 4 x ) - f ( x ) 4 x dy dx is also called ordinary derivative of y for a single-variable function, usually denoted by y 0 ( x ) . 1.2 Function of several variables, y = f ( x 1 , x 2 , · · · , x n ) . y is called dependent variable . x 1 , x 2 , · · · , x n are called independent variables . For a 3-variable function w = f ( x, y, z ) , the partial derivative of w with respect to x is deﬁned as ∂w ∂x = lim 4 x 0 f ( x + 4 x,y,z ) - f ( x,y,z ) 4 x , i.e., rate of change of w due to the change in x , with variables y, z ﬁxed. 1

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
2. Deﬁnition Ordinary Diﬀerential Equations (ODE) are equations contain ordinary derivatives only. Partial Diﬀerential Equations (PDE) are equations contain partial derivatives. # We will consider ODE only in this course! 3. Where are ODE come from when studying physical phenomena through mathematical modelling. physical phenomenon mathematical modelling ODE example of a simplest ODE free falling F = ma mg = m dv dt g = v 0 ( t ) v = ´ gdt + c = gt + c # Solving ODE always involves integration! In the more realistic case when air resistance (drag) is considered, the ODE for the free falling problem is more complicated. mathematical modelling of drag force, drag = γv (proportional to velocity), Newton’s law yields mg - γv = m dv dt dv dt = g - γ m v E.g. Pendulum v = tangential velocity, l = length of string, θ = angular displacement v = l dt , a = dv dt = l d 2 θ d 2 t Balance of force ma = - mgsinθ ml d 2 θ d 2 t = - mgsinθ θ 00 + g l sinθ = 0 (non-linear ODE) for small θ , θ 00 + g l θ = 0 (linear ODE) 2
E.g. Spring-mass system m = mass, x = displacement, k = spring constant s.t. the force of spring = kx from the Hook’s law. Balance of force gives

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

## This note was uploaded on 05/11/2011 for the course MATH 150 taught by Professor T.qian during the Spring '09 term at HKUST.

### Page1 / 15

math150L2_part1 - Section I - Introduction 1. Preliminaries...

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online