110年 - 110 國立臺灣大學_碩士班招生考試_部分系所:工程數學(H)#100779

1. Curves of constant temperature T(x, y)=const in a temperature field are called isotherms. Their orthogonal trajectories are the curves along which heat will flow (in regions that are free of sources or sinks of heat and are filled with a homogeneous medium). If the isotherms are given by ,what are the curves of heat flow? (1.5%)

2. Solve: (10%)

3. Find the Maclaurin series of (10%)

4. Let M_n is the nxn matrix with 3's on the main diagonal except the first entry which is 1, 1's directly above the main diagonal and (-1's directly below the main diagonal, and O's elsewhere. For example, We define D_n = det(M_n). Please (a) Find D₁, and D₂, (b) If n ≥ 3, find a formula for D_n in terms of, (c) Find a formula for D_n in terms of n. (15%)

(a) solve =0 at y = O and y = B;

6. are shown in the following figure, please prove that(10%) 

7. Show that if u = u(x,y) and ∇²u = 0 (Laplace's equation),
 then v = u(x² - y², 2xy) would also fulfill Laplace's equation.

8. Find the convolution  of