PDF of Lecture Notes - School of Mathematical Sciences
PDF of Lecture Notes - School of Mathematical Sciences
PDF of Lecture Notes - School of Mathematical Sciences
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∂<br />
∂r g 1(r, θ) = ∂ r cos θ = cos θ<br />
∂r<br />
∂<br />
(r cos θ) = −r sin θ<br />
∂θ<br />
∂<br />
∂r g 2(r, θ) = ∂ r sin θ = sin θ<br />
∂r<br />
∂<br />
r sin θ = r cos θ<br />
∂θ<br />
1. DISTRIBUTION THEORY<br />
⎡<br />
cos θ<br />
⇒ G = ⎣<br />
sin θ<br />
⎤<br />
−r sin θ<br />
⎦<br />
r cos θ<br />
⇒ det(G) = r cos 2 θ − (−r sin 2 θ)<br />
= r cos 2 θ + r sin 2 θ = r.<br />
Step 2<br />
Now apply theorem 1.8.4.<br />
Recall,<br />
f Z (z 1 , z 2 ) = 1 √<br />
2π<br />
e − 1 2 z2 1 ×<br />
1<br />
√<br />
2π<br />
e − 1 2 z2 2<br />
= 1<br />
2π e− 1 2 (z2 1 +z2 2 ) .<br />
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