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).