ME5331 Some useful tips
13.2 What are the definitions of continuity and differentiability ? Are there any functions that are not continuous but differentiable ?
13.4 (Numerical differentiation/integration) What precaution must one take when integrating a function numerically ? What kind of functions cause numerical integration to fail ?
13.5
  1. For a function to be expanded by the Taylor series, its Lagrange remainder must be convergent to 0. The function, exp (−1/x2), cannot be expanded by the Taylor series about x = 0 because its Lagrange remainder does not converge to 0.
  2. It is possible to get around the formula for the Taylor series by taking advantage of well-known series expansion such as the geometric series and binomial expansion etc ....
  3. Some of series worth remembering are:

    1. 1
      1 + x
      		 = 1 − x + x2 − x3 + x4 − …

    2. sin x = x − x3
      3!
      		 + x5
      5!
      		 − x7
      7!
      		 + …

    3. cos x = 1 − x2
      2!
      		 + x4
      4!
      		 − x6
      6!
      		 +…

    4. ex = 1 + x + x2
      2!
      		 + x3
      3!
      		 + x4
      4!
      		 + …

    5. ln (1+ x) = x − x2
      2
      		 + x3
      3
      		 − x4
      4
      		 + …

    6. (1+x)n = 1 + n x + n (n−1) x2
      2
      		 + n(n−1)(n−2) x3
      3!
      		 + …
  4. The Taylor formulas for single variable functions and multi-variable functions can be expressed in a unified format by using

    D = (x−a) d
    dx
    		     single variable
    or as

    D = (x − a) ∂
    ∂x
    		 +(y − b) ∂
    ∂y
    		     two variables
    as
    f(x) = f|x=a + Df|x=a + D2f
    2!
    		 |x=a + D3f
    3!
    		 |x=a +…

    f(x, y) = f|(x, y) = (a, b) + Df|(x, y)=(a, b) + D2f
    2!
    		 |(x, y)=(a, b) + D3f
    3!
    		 |(x, y)=(a, b) +…
3.6
  1. Try to understand the implicit function theorem for two variable functions (Jacobian) from the implicit function theorem for single variable functions.
  2. The Jacobian J(u, v)/J(x, y) is a scaling factor from the (x, y) coordinate system to the (u, v) coordinate system.
13.7 What is the advantage of the Lagrange multiplier method ?
13.8 How is the Leibniz rule used in physics ? When investigating the rate of time change, what's wrong with taking time derivatives alone ?
14.4 The vector triple product identity,
a ×(b ×c) = (a, c) b −(a, b) c
implies that the right hand side contains no cross products. Note that the triple product is always expressed as the difference between the vectors in the parenthesis (and the middle one comes first).
14.3 Can you compute the area of the parallelogram spanned by two independent vectors and the volume of the parallelepiped spanned by three independent vectors ?
15.4
a x + b y + c z = d represents a plane and the vector (a, b, c) is perpendicular to this plane. In general, a vector perpendicular to a given surface expressed by f(x, y, z)=0 can be obtained by taking the gradient of f.
15.5
Ru = ∂R/∂u and Rv = ∂R/∂v are base vectors on the curved surface and form a tangent plane. If the equation of a plane is expressed as z = f(x, y), it's faster to use
dS =
√
 
1 + fx2 + fy2
 
as an area element.


File translated from TEX by TTH, version 4.03.
On 02 Oct 2022, 15:08.