The line integral of \(F(x, y) = (x, y) \) along an ellipse \(x^2 + 2y^2 = 1 \) is
zero.
TRUE
FALSE answer± The curl \(Q_x - P_y \) of the vector field \(F(x, y) = (P, Q) \) is 0. By Green's theorem, the
line integral is zero. An other way to see this is that F is a gradient field \(F = \nabla f \) with
\(f (x, y) = (x^2 + y^2 )/2 \). Therefore \(F \) is conservative: the line integral along any closed curve in the plane is zero.
It is Hobbit time. The following regions resemble ancient runes of the Anglo
Saxons studied by JRR Tolkien. (A is "b", B is "d", C is "st", and D is "oe").
A,B,C,D
B,C,D,A
B,C,A,D
D,C,B,A answer± Set up all integrals.
The solid enclosed by the surfaces \(z= 2-\sqrt{x^2+y^2} \) and \(z=\sqrt{x^2+y^2} \) has the volume \( \int^{2\pi}_0 \int^1_0 \int^{2-r}_r r dzdrd\theta \)
TRUE
FALSE answer± Yes, this is the volume in cylindrical coordinates.
Match the solids with the triple integrals. Also here, there is an exact match:
B,C,A,D
A,C,D,B
A,B,C,D
D,A,B,C answer± All integrals are in cylindrical coordinates.
Assume (0,0) is a global maximum of \(f(x, y) \) on the disc \(D={x^2+y^2 \leq 1}\), then \(\iint_D f(x, y)dxdy \leq \pi f(0,0) \).
Match the following vector fields in spacewith the corresponding formulas:
IV,II,I,III
II,IV,III,I
IV,II,I,III
I,II,III,IV answer± First identify the first and the second picture, there the field has no z, and is constant respectively.
The figures display vector fields in the plane. There is an exact match.
II,I,IV,III
III,IV,II,I
I,II,IV,III
I,IV,II,III answer± The first vector field has zero \( x \), the second one is anti-radial, the fourth vector field has constant \( y\).
If the line integral of \(F\) along the closed loop \(x^2+y^2= 1, z= 0\) is zero then the vector field is conservative.
TRUE
FALSE answer± It needs to be with respect to all paths.
Let \(F \) be a vector field which coincides with the unit normal vector \(N \) for each point on a curve \(C\). Then \(\int_C F\cdot dr = 0\).
TRUE
FALSE answer± The vector field is orthogonal to the tangent vector to the curve.
If \(F(x, y, z) = \lbrace 1,1,1 \rbrace \), and \(C \) is a curve, then \( \int_C F \cdot dr \) is the arc length of \(C\).
TRUE
FALSE answer± If we write down the line integral we get \(\int^b_a P(r(t)) +Q(r(t)) +R(r(t))dt\) and not \(\int^b_a \sqrt{P(r(t))^2+Q(r(t))^2+R(r(t))^2}dt \).