1. State the inverse function theorem and give a proof.
2. State the Fubini theorem and give a proof.
3. Calculate the volume of B^4 = { x屬於 R^4 | ∥x∥< 1 }
4. Calculate
∫ log√x^2+y^2
─────── dxdy.
R^2 (1+x^2+y^2)^2
5. Let u = x + y - εxy + x^4
v = x - y + εxy + y^4, 0 < ε << 1.
Prove that x, y can be expressed in terms of u, v near the origin.
Let A be a 2 ×2 matrix, B, C, D 屬於 R^2 such that
( x ) = A(u) + ε(Bu^2 + Cuv + Dv^2)
( y ) (v)
Find A, B, C, D.
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