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Unit 6 - Practice Quiz

MTH166

1 Which of the following defines the work done by a force field along a curve ?

A.
B.
C.
D.

2 A vector field is conservative if:

A.
B.
C.
D. is constant

3 If is a conservative vector field, then along a closed curve is:

A.
B. $1$
C. $0$
D. Dependent on the area enclosed

4 Green's Theorem connects a line integral along a simple closed curve to a:

A. Surface integral over a closed surface
B. Volume integral over the region bounded by
C. Double integral over the plane region bounded by
D. Line integral along a different curve

5 According to Green's Theorem, is equal to:

A.
B.
C.
D.

6 Which of the following formulas calculates the area of a region bounded by a closed curve using Green's Theorem?

A.
B.
C.
D. All of the above

7 In Green's Theorem, the curve must be traversed in which direction for the theorem to hold directly?

A. Clockwise
B. Counter-clockwise (Positive orientation)
C. Any direction
D. Upwards

8 The value of where is the line segment from to is:

A. $1$
B. $0.5$
C. $0$
D. $2$

9 A surface integral represents:

A. The volume enclosed by the surface
B. The flux of across the surface
C. The circulation of around the boundary
D. The surface area of

10 For a surface defined by , the element of surface area is given by:

A.
B.
C.
D.

11 Stokes' Theorem relates a surface integral of the curl of a vector field to:

A. A volume integral of the divergence
B. A line integral of the vector field around the boundary curve
C. A double integral over the region
D. The scalar potential of the field

12 Mathematically, Stokes' Theorem is expressed as:

A.
B.
C.
D.

13 Gauss's Divergence Theorem relates a surface integral over a closed surface to:

A. A line integral along the boundary
B. A volume integral over the enclosed region
C. A surface integral over a different surface
D. The curl of the vector field

14 The mathematical statement of Gauss's Divergence Theorem is:

A.
B.
C.
D.

15 If , then is:

A. $0$
B. $1$
C. $3$
D.

16 Using the Divergence Theorem, the flux of through a closed surface enclosing a volume is:

A.
B.
C.
D. $0$

17 If is a closed surface enclosing a volume and is a solenoidal vector field, then the flux is:

A. $1$
B.
C.
D. $0$

18 Which theorem is a special case of Stokes' Theorem applied to the -plane?

A. Gauss's Divergence Theorem
B. Green's Theorem
C. Fundamental Theorem of Calculus
D. Euler's Theorem

19 In Stokes' Theorem, the direction of the unit normal vector is determined by:

A. The Left-Hand Rule
B. The Right-Hand Rule relative to the path traversal
C. The direction of the z-axis always
D. Arbitrary selection

20 The line integral along the circle traversed counter-clockwise is:

A. $0$
B.
C.
D.

21 If , then equals:

A.
B.
C.
D.

22 If a vector field is path independent, then:

A. It is not conservative
B.
C. can be written as
D. Work done in a closed path is non-zero

23 The surface integral represents:

A. Volume of the solid
B. Mass of the surface
C. Area of the surface
D. Moment of inertia

24 For a surface , the unit normal vector is given by:

A.
B.
C.
D.

25 In the evaluation of , if the surface is projected onto the -plane (region ), then is replaced by:

A.
B.
C.
D.

26 Evaluate where is the curve from to .

A.
B.
C.
D.

27 Which theorem would you use to convert a surface integral over a closed surface into a volume integral?

A. Green's Theorem
B. Stokes' Theorem
C. Gauss's Divergence Theorem
D. Pythagoras Theorem

28 Which theorem would you use to convert a line integral over a closed curve to a surface integral?

A. Green's Theorem
B. Stokes' Theorem
C. Gauss's Divergence Theorem
D. Fundamental Theorem of Line Integrals

29 If everywhere in a region bounded by , then the net flux through is:

A. Infinite
B. Zero
C. Equal to the volume
D. Equal to the surface area

30 The parametric equations , , represent:

A. A cylinder
B. A cone
C. A sphere
D. A plane

31 If , calculate the curl .

A.
B.
C.
D.

32 The region enclosed by the path in Green's theorem must be:

A. Simply connected
B. Three dimensional
C. Infinite
D. Discontinuous

33 Evaluate around the square bounded by .

A. $0$
B. $1$
C.
D. $2$

34 The vector area element is defined as:

A.
B.
C.
D.

35 For a plane surface (the -plane), the unit normal vector is:

A.
B.
C.
D.

36 If represents the velocity of a fluid, represents:

A. Flux
B. Circulation
C. Divergence
D. Gradient

37 Gauss's Divergence theorem requires the surface to be:

A. Open
B. Closed
C. Flat
D. Circular

38 In the formula , if is the outward normal, a positive result implies:

A. Net flow into the volume
B. Net flow out of the volume
C. Zero flow
D. Rotational flow

39 If , then is:

A.
B. $3$
C.
D. $1$

40 Line integrals are independent of the parameterization of the curve, provided:

A. The orientation is reversed
B. The orientation is preserved
C. The curve is closed
D. The field is zero

41 Applying Stokes' Theorem to a surface which is a closed surface (like a sphere) results in:

A. The volume
B. Zero
C. The surface area
D. Infinite

42 The scalar product in Cartesian coordinates is:

A.
B.
C.
D.

43 Find over the plane for .

A. $1$
B. $2$
C. $4$
D. $0$

44 Which of the following is required for to be independent of path?

A. must be constant
B. The region must be multiply connected
C. in a simply connected region
D.

45 Green's Theorem in the plane can be written in vector notation as:

A.
B.
C.
D. None of the above

46 If (a constant vector), then over a closed surface is:

A.
B. $0$
C. Volume
D. Undefined

47 What is the physical meaning of at a point?

A. The rotation of the field at that point
B. The rate of change of the field along a line
C. The flux density (outflow per unit volume)
D. The potential energy

48 If , what is ?

A.
B.
C. $0$
D.

49 When computing surface area of a sphere , the limits for in spherical coordinates are:

A. $0$ to
B. $0$ to
C. to
D. $0$ to

50 Which identity relates the Divergence Theorem and Green's First Identity?

A.
B.
C.
D.