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Finals Notes

Mathematics Notes > Calculus III Notes

This is an extract of our Finals document, which we sell as part of our Calculus III Notes collection written by the top tier of Nanyang Technological University students.

The following is a more accessble plain text extract of the PDF sample above, taken from our Calculus III Notes. Due to the challenges of extracting text from PDFs, it will have odd formatting:

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 =


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

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




f (x, y, z)dxdydz =

V [?][?][?]

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

W is the same region in cylindrical coordinates.


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