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Lecture notes on the change of variables technique in multivariable calculus. It covers the transformation of coordinates, focusing on the jacobian determinant and its role in calculating areas under mappings. The notes include examples of linear and non-linear mappings, such as polar coordinates, and demonstrate how to find the image of geometric shapes under these transformations. The document also explains how to compute the jacobian determinant for linear maps and its relation to the area of parallelograms, offering a comprehensive overview of the topic for students studying multivariable calculus. It is useful for understanding coordinate transformations and their applications in integration and area calculations. (438 characters)
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changeofilariables g
skipping The phrase change^ of (^) variables means changing coordinates Here we^ will^ learn
dA dxdy rdrd dV dxdydz rolzdrolf psinddpolbd 0 MITEE.lt we (^) will consider
i.e (^) functions G (^) D IR (^) on a (^) domain D R We will denote
in (^) D as nu and (^) points in^ B^ as^ x^ y GCD
So x u^ v y you^ v (^) Glu 4 x^ use y u (^) D Example We (^) have alreadysecretly^ been thinking about this^ Consider G (^) 0,0 (^) 0,2 7 R G r^ rcos (^) rsin This is^ the^ map that^ translates polar cords into (^) rectangular coordinates Another (^) classic type of
linearnaps G u v^ AU Cv^ But (^) Dv A^ B^ C^ D^ constant
have taken linear^ algebra notice we^ can describe 6 with^ a (^) matrix B 5 Y^ E
Linear mappings send straight lines (^) to straight lines parallel lines to^ parallel lines
We have uv y ux a G c^ x^ Cv^ ex^ x
So (^) x CV (^1) V So y CV c^ E
Glu c^ UC^ uc^ x^ y UC (^) U (^) XC y
y xc since^ us0 xoo b
Hence (^) the image of 1,2 1,2^ consists^ of x y such^ that y x y 4x or in^ other words
(^4 1 )
Inverse
y UV xx (^) v V2^ v V (^) E since^11 So (^) u xv (^) xFE P So G^ x^
y F F Leachantoramapping
Jacindetrminant (^) of G u^ v^ u^ v^ y
sacco 3 1 This is^ often^ denoted Note Jac^
function (^) of a v Then The^ Jacobian
Gcu x (^) Au Cv But^ Dv (^) is (^) constant with value JacCO AD^ BC^ B 5 And (^) for any domain^ D area GCD^ Jac^ G (^) area D
