Math Lecture 1

2. Math 304
Multivariable Calculus & Differential Equations
Wafik Lotfallah

3. Course Material
T1) Thomas' Calculus, 11th edition, by M.D. Weir, J. Hass,
and F.R. Giordano, Pearson Add, 2005
T2) Calculus, 5th edition, by J. Stewart, Brooks/Cole, 2003
T3) Advanced Engineering Mathematics, 8th edition, by E.
Kreyszig, J. Wiley & Sons, 1999
T4) Shaum’s Outline of Advanced Mathematics for
Engineers and Scientists, 1st edition, by M.R. Spiegel,
McGraw-Hill, 1971
T5) Supplementary Material on the inter/intranet.
Course website: http://math.guc.edu.eg/math304/

4. Course Assessment System
 Homework + Classwork: 10%
 Quizzes: 25%, both counted (NO make-ups)
 Midterm: (35%)
 Final: (45%, comprehensive)

5. Prerequisites: Math 103, Math 203.
However, apart from being able to differentiate
and integrate, we will review anything you
need to know for the course.

6. Lecture 1
Revision on Vectors

7.  Find and interpret
 the sum of two vectors
 the product of a vector by a scalar
 the dot product of two vectors
 the magnitude of a vector
 the unit vector in a specified direction
 the angle between two vectors
 the cross product of two vectors
 the work done by a constant force in moving objects
along a straight line segment
 Find and graph equations of
 spheres
 Planes
 Straight lines
Lecture Objectives

8. The 3 Dimensional Space

9. The 3 Coordinate Planes

10. Points in Space

11. Distances

12. Spheres

13. Vectors

14. Equality of
Vectors

15. Vectors in
Standard
Position
Notation:
v = v1, v2, v3
= x2 x1, y2 y1, z2 z1

16. Length of
Vectors

19. Subtraction:

21. An Alternative Representation:
Letting i = 1, 0, 0; j = 0, 1, 0; k = 0, 0, 1;
we can write:
v = x, y, z = x 1, 0, 0 + y 0, 1, 0 + z 0, 0, 1
= x i + y j + z k
Example: 1, 2, 3 = i + 2 j + 3 k
2, 1, 0 = 2 i  j

22. Properties of the Norm: Let u and v be vectors and c a
scalar. Then:
 || v || = 0 iff v = 0
 || cv || = |c| || v ||

23. Vector Direction:
Example: Find a vector of length 5 in the
same direction of the vector v = 2, 1, 2.

24. The Dot Product

25. Equivalent Definition:
uv = |u| |v| cos

26. Orthogonal (Perpendicular)
Vectors:
Letting  = /2 in uv = |u||v|cos():
Example: Show that 1, 2, 3 and 2, 1, 0
are orthogonal.

27. Work

28. The Cross Product

29. An Alternative Definition:

31. Also note:
6. u  u = 0

32. Equations of Planes

34. Example: Find the equation of the plane through the
origin (0, 0, 0) and perpendicular to 2, 0, 5.
Solution: 2(x  0) + 0(y  0) + (5)(z  0) = 0
2x  5z = 0

35. Example: Find the equation of the plane through
the points P(1, 0, 0); Q(0, 2, 0); R(0, 0, 3).
Solution 1: First find a normal vector by taking
the cross product:
Then use the plane formula:
6(x  1) + 3(y  0) + 2(z  0) = 0
or 6x + 3y + 2z = 6
k
j
i
k
j
i
2
3
6
3
0
1
0
2
1
3
,
0
,
1
0
,
2
,
1 









PR
PQ

36. Example: Find the equation of the plane through
the points P(1, 0, 0); Q(0, 2, 0); R(0, 0, 3).
Solution 2: Start with Ax + By + Cz = D, and
substitute the points
P(1, 0, 0); Q(0, 2, 0); R(0, 0, 3) to get:
A = D; 2B = D; 3C = D
Letting D = 1, we get A = 1, B = 1/2, C = 1/3, so
the equation becomes: x + (1/2)y + (1/3)z = 1,
or 6x + 3y + 2z = 6

37. Equations of Straight Lines

39. Symmetric Equations for a Line: Eliminate the parameter t
3
0
2
0
1
0
v
z
z
v
y
y
v
x
x
t







42. Thank you for listening.
Wafik