# Key Points of MA 1505 Mathematics I

Key Point of Final Exam on MA 1505: 2 hours, 8 questions, each questions contains two small questions. i.e. 16 questions on 2 hours.

There are four parts before mid-term exam.

1. Derivatives.

MA 1505 Tutorial 1: Derivative

Calculate the derivatives of functions and remember the rules of derivatives, such as

$(f \pm g)^{'}(x)= f^{'}(x) \pm g^{'}(x)$

$( f \cdot g)^{'}(x)= f^{'}(x) \cdot g(x) + f(x) \cdot g^{'}(x)$

$( \frac{f}{g})^{'}(x) = \frac{ f^{'}(x) \cdot g(x) - f(x) \cdot g^{'}(x) }{ g^{2}(x) }$

$( f\circ g)^{'}(x) = f^{'}(g(x)) \cdot g^{'}(x)$

2. Integration.

MA 1505 Tutorial 2: Integration

The method of integration by parts.

3. Power Series and Taylor Series.

MA 1505 Tutorial 3: Taylor Series

Definition of Taylor series.

4. Analytic Geometry.

(1) Equations of planes and lines in $\mathbb{R}^{3}.$

(2) Dot Product:

Assume $\textbf{a}=(a_{1}, a_{2}, ... , a_{n})$ and $\textbf{b}=(b_{1}, b_{2}, ... , b_{n})$ are vectors in $\mathbb{R}^{n}$, the dot product of a and b is

$\textbf{a} \cdot \textbf{b} = \sum_{i=1}^{n} a_{i} \cdot b_{i} = || \textbf{a} || \cdot || \textbf{b} || \cdot \cos \theta,$

where ||a|| denotes the length of the vector a||b|| denotes the length of the vector b and $\theta$ denotes the angle between vectors a and b.

(3) Cross Product:

Assume a and b are two vectors in $\mathbb{R}^{3}.$ The cross product is defined by the formula

$\textbf{a} \times \textbf{b} = || \textbf{a} || \cdot || \textbf{b} || \cdot \sin \theta \cdot \textbf{n},$

where θ is the measure of the smaller angle between a and b (0° ≤ θ ≤ 180°), ‖a‖ and ‖b‖ are the length of vectors a and b, and n is a unit vector perpendicular to the plane containing a and b in the direction given by the right-hand rule as illustrated.

These are after mid-term exam.

5. Fourier Series of Periodic Functions.

MA 1505 Tutorial 5: Fourier Series

(1) Calculate the Fourier series of periodic functions with period $2 \pi$ or $2 L (L \neq \pi).$

For period $2 \pi,$ the fourier series of $f(x)$ is

$a_{0} + \sum_{n=1}^{\infty} (a_{n} \cos (nx) + b_{n} \sin (nx) ).$

(2) Pay attention to how to calculate the summation of the Fourier coefficients $\sum_{n=0}^{\infty} a_{n}$ and $\sum_{n=1}^{\infty} a_{n}.$

6. Multiple Variable Functions.

MA 1505 Tutorial 6: Partial Derivatives and Directional Derivative

(1) Calculate the partial derivatives and the gradient field of two variables function $z=f(x,y)$.

(2) Directional derivatives of $z=f(x,y)$ on some point $(x_{0}, y_{0})$ under some unit vector $u= a \textbf{i} + b \textbf{j}$ with $a^{2}+b^{2}=1,$ where i=(1,0), j=(0,1).

(3) If $f(x,y)$ has continuous secondary partial derivatives, then $f_{xy}= f_{yx}.$

(4) The critical points of $z=f(x,y),$ and check the property of the critical points: Saddle PointsLocal Maximum or Local Minimum.

(5) Langrange’s Method: calculate the maximum value of some function under some condition.

Example 1: Calculate the minimum and maximum value of $f(x,y)=xy$ under the condition $x^{2}+y^{2}=1.$

Example 2: Calculate the maximum value of $f(x,y)=xy$ under the condition $x+y=1, x, y \geq 0.$

7. Multiple Integration.

MA 1505 Tutorial 7: Integration of Two Variables Functions

MA 1505 Tutorial 8: Surface Area and Volume

(1) Polar Coordinate:

$x= r \cos \theta \text{ and } y=r \sin \theta.$

$\iint_{R} f(x,y) dxdy = \iint_{D} f(r \cos \theta , r \sin \theta) \cdot r dr d\theta.$

(2) Reverse the order of integration: Find the region $R.$

(3) The Volume of some solid:

If $z=z(x,y) \geq 0$ where $(x,y) \in R$,  the volume is $\iint_{R} z(x,y) dxdy.$

Example: the volume of the sphere with radius $R.$

(4) The area of some surface:

If the equation of the surface is $z=z(x,y),$ where $(x,y)\in R,$ the area of the surface is $\iint_{R} \sqrt{ 1+ z_{x}^{2}+ z_{y}^{2}} dxdy.$

Example: the surface area of the sphere with radius $R.$

8. Line integrals and Green’s Theorem.

MA 1505 Tutorial 9 and 10: Line Integration and Green’s Formula

(1) Line integral of a scalar field:

(2) Line integral of a vector field:

(3) Conservative vector field:

The vector field is called a conservative vector field, i.e. there exists a scalar function f such that $\nabla f= \textbf{F}.$

(4) Newton-Leibniz formula on gradient field:

$\int_{C} \nabla f \cdot d \textbf{r} = f( \textbf{r}(b)) - f( \textbf{r} (a)),$ where C is a curve, $r: [a,b] \rightarrow C.$

(5) Green’s Theorem: The relationship between line integrals of vector fields and double integrations.

$\oint_{\partial D} P dx + Q dy= \iint_{D} ( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} ) dxdy.$

9. Surface Integrals

MA 1505 Tutorial 11: Surface Integral, Divergence Theorem and Stokes Theorem

(1) Surface Integrals of Scalar Fields:

(2) Surface Integrals of Vector Fields:

(3) Divergence Theorem: The relationship between surface integrals of vector fields and triple integrations.

(4) Stokes Theorem: The relationship between line integrals of vector fields and surface integrals of vector fields.