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Final Exam

Part A. Multiple Choice Questions (5 points each, total 25 points)

1.In three-dimensional space, the dot product of the vectors a=(2,−3,5) and b=(1,4,−2) is:

●     A) 0

●     B) 10

●     C) -1

●     D) 1

2.If r(t)=(t,t2 ,t3), then drdt is:

●     A) (1,2t,3t2)

●      B) (0,0,0)

●     C) (t,2t,3t)

●     D) (1, 1, 1)

3.If the function f(x,y)=x2+y2, then the gradient ▽fat the point (1, 1) is:

●     A) (2,2)

●     B) (1, 1)

●     C) (2, 1)

●     D) (1,2)

4. For the double integral over the region D defined by 0≤x≤1 and 0≤y≤1−x, the value of is:

● A) 6/1

● B) 4/1

● C) 3/1

●    D) 2/1

5.According to Stokes' Theorem, where C is:

●     A) An open curve

●      B) A closed curve

●     C) A plane

●      D) A solid

Part A. Fill-in-the-Blank Questions (5 points each, total 25 points)

6.In three-dimensional space, the cross product

a=(2,2, 1) and b=(1,0,3) is a×b= .

7.Let f(x,y)=x3y+2xy2, then the partial derivative ∂f∂x at the point (1,2) is .

8.The double integral over the region D represents .

9.The line integral represents .

10.If the divergence of a vector field is zero, then the field is .

Part C. Short Answer Questions (10 points each, total 50 points)

11.Please explain the geometric meaning of the partial derivative of a multivariable function, and provide an example.

12.Use double integrals to compute the integral over the region D defined by 0≤x≤1 and 0≤y≤1−x.

13.Briefly describe Green's Theorem and its applications in physics.

14.State and prove the Divergence Theorem.

15.Calculate the line integral F(x,y)=(y,x) along the straight line segment from (0,0) to (1, 1).