Formula Sheet Math Multi-Variable Calculus

University of Babylon
College of Engineering
Mechanical Engineering Dept.
Subject : Mathematics 4
Class: 2nd
Date: / / 2016

Formula Sheet Math Multi-Variable Calculus


1. length of a vector in Space ?v? = ? v12 + v22 + v32

2. 2 dimensional dot product u?v = u1v1 + u2v2

3. 3 dimensional dot product u?v = u1v1 + u2v2 +u3v3

4. Angle between two vectors cos ? = __u?v__
?u ? ?v?

5. Cross product u x v = (u2v3 – u3v2)i – (u1v3 – u3v1)j + (u1v2 - u2v1)k

6. parametric form equations of a line in space x = x1 + at
y = y1 + bt
z = z1 + ct

7. symmetric form of the equations of a line in space
x-x1 = y – y1 = z – z1
a b c

8 Standard equation of a plane in Space

a(x-x1) + b(y-y1) + c (z – z1) = 0

9. general form of the equation of a plane in Space ax +by +cz +d = 0

10. cylindrical to Cartesian (rectangular):
x = r cos ? y = r sin ? z = z

11. Cartesian (rectangular) to cylindrical
r2 = x2 + y2 tan ? = y/x z=z

14. total differential: dw = ?w dx + ?w dy + ?w dz + ?w du
?x ?y ?z ?u


15 Chain rule one independent variable dw = ?w ?x + ?w ?y
dt ?x ?t ?y ?t

16. Chain rule two independent variables

?w = ?w ?x + ?w ?y and ?w = ?w ?x + ?w ?y
?s ?x ?s ?y ?s ?t ?x ?t ?y ?t

17. Chain rule Implicit differentiation

dy = - Fx(x,y)
dx Fy (x,y) Fy(x,y) ? 0

18. Chain rule Implicit differentiation

dz = - Fx(x,y,z) dz = - Fy(x,y,z)
dx Fz(x,y,z) dy Fz(x,y,z) Fz(x,y,z) ? 0

19. Directional Derivative

For unit vector u=cos ? i + sin ? j

Duf (x,y) = fx(x,y) cos ? + fy(x,y) sin ?

20. Gradient of f
f(x,y) = fx(x,y)i +fy(x,y)j

21. Second Partials Test
f must have continuous second derivatives on an open region containing point (a,b) for which
fx (a,b) =0 fy(a,b) = 0

To test for extrema consider the quantity:

D= fxx(a,b) fyy(a,b) – [fxy(a,b)]2

1. if d > 0 and fxx(a,b) > 0, then f has a relative minimum at (a,b)
2. if d > 0 and fxx(a,b) < 0, then f has a relative maximum at (a,b)
3. if d < 0 then (a,b,f(a,b)) is a saddle point.
4. the test is inconclusive if d = 0

22. Ellipse x2/a2 + y2/b2 = 1

23. Ellipsoid x2/a2 + y2/b2 +z2/c2 =1

24. Hyperbola x2/a2 - y2/b2 = 1

25. Hyperboloid of one sheet x2/a2 + y2/b2 – z2/c2 = 1

26. Hyperboloid of two sheets x2/a2 - y2/b2 – z2/c2 = 1

27. Elliptic cone x2/a2 + y2/b2 – z2/c2 = 0

28. Elliptic Paraboloid z = x2/a2 +y2/b2

29. Hyperbolic paraboloid z = y2/b2 –x2/a2

30.

31.


32.

33. Using gradients to compute directional derivatives:

34

35. LaGrange’s Theorem