Derivative of cos(b)+log(tan(b/2)) wrt b assuming log is ln:
-sin(b)+½cot(b/2)sec²(b/2)=
-sin(b)+½(cos(b/2)/sin(b/2))/cos²(b/2)=
-sin(b)+1/(2sin(b/2)cos(b/2))=
-sin(b)+1/sin(b)=(1-sin²(b))/sin(b)=cos²(b)csc(b) or cos(b)cot(b).
n=∂φ/∂a × ∂φ∂b. (Vector cross product)
∂φ/∂a=(-sin(a)cos(b),cos(a)sin(b),0)
∂φ∂b=(-cos(a)sin(b),sin(a)cos(b),cos²(b)csc(b)),
n=(cos(a)cos²(b),sin(a)cos²(b)cot(b),-sin²(a)cos²(b)+cos²(a)sin²(b)).
|n|=√(cos²(a)cos⁴(b)+sin²(a)cos⁴(b)cot²(b)+sin⁴(a)cos⁴(b)-2sin²(a)cos²(b)cos²(a)sin²(b)+cos⁴(a)sin⁴(b)).
Let A=sin²(a), B=sin²(b), then 1-A=cos²(a), 1-B=cos²(b), cot²(b)=(1-B)/B=1/B-1.
|n|=√[(1-A)(1-B)²+A(1-B)²(1/B-1)+A²(1-B)²-2AB(1-A)(1-B)+(1-A)²B²].
|n|=√[1-2B+B²-A+2AB-AB²+A/B-2A+3AB-A-AB²
+A²-2A²B+A²B²-2AB+2A²B+2AB²-2A²B²+B²-2AB²+A²B²].
|n|=√[1-2B+2B²-4A-2AB²+A/B+3AB+A²].
|n|=√[1-2B+B²+A²+2AB+B²+AB-4A-2AB²+A/B].
|n|=√[(1-B)²+(A+B)²+A(B-4-2B²+1/B)].
|n|=√[cos⁴(b)+(sin²(a)+sin²(b))²+sin²(a)(sin²(b)-4-2sin⁴(b)+csc²(b))].
Unit normal vector=n/|n|.
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