Mathematics Notes > Calculus III Notes

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Calculus

Andre Sostar 2014

Contents 1 Substitution in multiple integrals

3 Polar coordinates

3 Cylindrical coordinates

3 Spherical coordinates

4 2 Vector elds and line integrals

5 Dot and Cross product of a vector eld

5 Gradient of a function

5 Directional derivative, Divergence, Curl

5 3 Line integral

6 Line integral

6 Work of a vector eld

6 4 Newton-Leibniz' and Green's Theorems

7 The Newton-Leibniz' Theorem

7 Conservative vector elds

7 Green's Theorem

8 1 5 Parametrization

9 Parametrization

9 2 1 Substitution in multiple integrals

Polar coodrinates:

r, th are dened as:

x = r cos th, y = r sin th

r2 = x2 + y 2 , r = r cos2 th + r sin2 th

dA = r * drdth

Finally where

[?][?]

f (x, y)dxdy =

D

[?][?]

r * f (r cos th, r sin th) drdth

E

E is the same region in r, th.

Cylindrical coordinates:

r, th, z are dened as:

x = r cos th, y = r sin th, z = z r2 = r2 cos2 th + r2 sin2 th

dV = rdr dthdz

Finally

where

[?][?][?]

f (x, y, z)dxdydz =

V [?][?][?]

r * f (r cos th, r sin th, y) drdthdz

W

W is the same region in cylindrical coordinates.

3

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