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.
Calculate the derivatives of functions and remember the rules of derivatives, such as
The method of integration by parts.
3. Power Series and Taylor Series.
Definition of Taylor series.
4. Analytic Geometry.
(1) Equations of planes and lines in
(2) Dot Product:
Assume and are vectors in , the dot product of a and b is
where ||a|| denotes the length of the vector a, ||b|| denotes the length of the vector b and denotes the angle between vectors a and b.
(3) Cross Product:
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.
(1) Calculate the Fourier series of periodic functions with period or
For period the fourier series of is
(2) Pay attention to how to calculate the summation of the Fourier coefficients and
6. Multiple Variable Functions.
(1) Calculate the partial derivatives and the gradient field of two variables function .
(2) Directional derivatives of on some point under some unit vector with where i=(1,0), j=(0,1).
(3) If has continuous secondary partial derivatives, then
(4) The critical points of and check the property of the critical points: Saddle Points, Local 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 under the condition
Example 2: Calculate the maximum value of under the condition
7. Multiple Integration.
(1) Polar Coordinate:
(2) Reverse the order of integration: Find the region
(3) The Volume of some solid:
If where , the volume is
Example: the volume of the sphere with radius
(4) The area of some surface:
If the equation of the surface is where the area of the surface is
Example: the surface area of the sphere with radius
8. Line integrals and Green’s Theorem.
(1) Line integral of a scalar field:
(2) Line integral of a vector field:
(3) Conservative vector field:
The vector field F is called a conservative vector field, i.e. there exists a scalar function f such that
(4) Newton-Leibniz formula on gradient field:
where C is a curve,
(5) Green’s Theorem: The relationship between line integrals of vector fields and double integrations.
9. Surface Integrals
(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.