5.2: Exercises

1 ✳

True or false?

1. \(\vecs{ \nabla} \cdot(\textbf{a} \times\vecs{r} ) = 0\text{,}\) where \(\textbf{a}\) is a constant vector in \(\mathbb{R}^3\), and \(\vecs{r} \) is the vector field \(\vecs{r} = (x, y, z)\text{.}\)
2. \(\vecs{ \nabla} \times(\vecs{ \nabla} f) = 0\) for all scalar fields \(f\) on \(\mathbb{R}^3\) with continuous second partial derivatives.
3. \(\vecs{ \nabla} \cdot(f \vecs{F} ) = \vecs{ \nabla} (f)\cdot \vecs{F} + f \vecs{ \nabla} \cdot\vecs{F} \text{,}\) for every vector field \(\vecs{F} \) in \(\mathbb{R}^3\) with continuous partial derivatives, and every scalar function \(f\) in \(\mathbb{R}^3\) with continuous partial derivatives.
4. Suppose \(\vecs{F} \) is a vector field with continuous partial derivatives in the region \(D\text{,}\) where \(D\) is \(\mathbb{R}^3\) without the origin. If \(\vecs{ \nabla} \cdot\vecs{F} \gt 0\) throughout \(D\text{,}\) then the flux of \(F\) through the sphere of radius \(5\) with center at the origin is positive.
5. If a vector field \(\vecs{F} \) is defined and has continuous partial derivatives everywhere in \(\mathbb{R}^3\text{,}\) and it satisfies \(\vecs{ \nabla} \cdot\vecs{F} = 0\text{,}\) everywhere, then, for every sphere, the flux out of one hemisphere is equal to the flux into the opposite hemisphere.
6. If \(\vecs{r} (t)\) is a twice continuously differentiable path in \(\mathbb{R}^2\) with constant curvature \(\kappa\text{,}\) then \(\vecs{r} (t)\) parametrizes part of a circle of radius \(1/\kappa\text{.}\)
7. The vector field \(\vecs{F} = \left( -\frac{y}{x^2+y^2}\,,\,\frac{x}{x^2+y^2}\right)\) is conservative in its domain, which is \(\mathbb{R}^2\text{,}\) without the origin.
8. If a vector field \(\vecs{F} = (P, Q)\) in \(\mathbb{R}^2\) has \(Q = 0\) everywhere in \(\mathbb{R}^2\text{,}\) then the line integral \(\oint\vecs{F} \cdot\text{d}\vecs{r} \) is zero, for every simple closed curve in \(\mathbb{R}^2\text{.}\)
9. If the acceleration and the speed of a moving particle in \(\mathbb{R}^3\) are constant, then the motion is taking place along a spiral.

2 ✳

True or false?

1. \(\vecs{ \nabla} \times(\textbf{a} \times\vecs{r} ) = 0\text{,}\) where \(\textbf{a}\) is a constant vector in \(\mathbb{R}^3\), and \(\vecs{r} \) is the vector field \(\vecs{r} = (x, y, z)\text{.}\)
2. \(\vecs{ \nabla} \cdot(\vecs{ \nabla} f) = 0\) for all scalar fields \(f\) on \(\mathbb{R}^3\) with continuous second partial derivatives.
3. \(\vecs{ \nabla} (\vecs{ \nabla} \cdot \vecs{F} ) = 0\) for every vector field \(\vecs{F} \) on \(\mathbb{R}^3\) with continuous second partial derivatives.
4. Suppose \(\vecs{F} \) is a vector field with continuous partial derivatives in the region \(D\text{,}\) where \(D\) is \(\mathbb{R}^3\) without the origin. If \(\vecs{ \nabla} \cdot\vecs{F} = 0\text{,}\) then the flux of \(\vecs{F} \) through the sphere of radius \(5\) with center at the origin is \(0\text{.}\)
5. Suppose \(\vecs{F} \) is a vector field with continuous partial derivatives in the region \(D\text{,}\) where \(D\) is \(\mathbb{R}^3\) without the origin. If \(\vecs{ \nabla} \times\vecs{F} =\vecs{0}\) then \(\oint_C\vecs{F} \cdot\text{d}\vecs{r} \) is zero, for every simple and smooth closed curve \(C\) in \(\mathbb{R}^3\) which avoids the origin.
6. If a vector field \(\vecs{F} \) is defined and has continuous partial derivatives everywhere in \(\mathbb{R}^3\text{,}\) and it satisfies \(\vecs{ \nabla} \cdot\vecs{F} \gt 0\text{,}\) everywhere, then, for every sphere, the flux out of one hemisphere is larger than the flux into the opposite hemisphere.
7. If \(\vecs{r} (t)\) is a path in \(\mathbb{R}^3\) with constant curvature \(\kappa\text{,}\) then \(\vecs{r} (t)\) parametrizes part of a circle of radius \(1/\kappa\text{.}\)
8. The vector field \(\vecs{F} = \left( -\frac{y}{x^2+y^2}\,,\,\frac{x}{x^2+y^2} \,,\,z\right)\) is conservative in its domain, which is \(\mathbb{R}^3\text{,}\) without the \(z\)-axis.
9. If all flow lines of a vector field in \(\mathbb{R}^3\) are parallel to the \(z\)-axis, then the circulation of the vector field around every closed curve is \(0\text{.}\)
10. If the speed of a moving particle is constant, then its acceleration is orthogonal to its velocity.

3 ✳

1. True or false? If \(\vecs{r} (t)\) is the position at time \(t\) of an object moving in \(\mathbb{R}^3\text{,}\) and \(\vecs{r} (t)\) is twice differentiable, then \(|\vecs{r} ''(t)|\) is the tangential component of its acceleration.
2. Let \(\vecs{r} (t)\) is a smooth curve in \(\mathbb{R}^3\) with unit tangent, normal and binormal vectors \(\hat{\textbf{T}}(t)\text{,}\) \(\hat{\textbf{N}}(t)\text{,}\) \(\hat{\textbf{B}}(t)\text{.}\) Which two of these vectors span the plane normal to the curve at \(\vecs{r} (t)\text{?}\)
3. True or false? If \(\vecs{F} = P\hat{\pmb{\imath}} + Q\hat{\pmb{\jmath}} + R\hat{\mathbf{k}}\) is a vector field on \(\mathbb{R}^3\) such that \(P\text{,}\) \(Q\text{,}\) \(R\) have continuous first order derivatives, and if \(\vecs{ \nabla} \times\vecs{F} = \vecs{0}\) everywhere on \(\mathbb{R}^3\), then \(\vecs{F} \) is conservative.
4. True or false? If \(\vecs{F} = P\hat{\pmb{\imath}} + Q\hat{\pmb{\jmath}} + R\hat{\mathbf{k}}\) is a vector field on \(\mathbb{R}^3\) such that \(P\text{,}\) \(Q\text{,}\) \(R\) have continuous second order derivatives, then \(\vecs{ \nabla} \times(\vecs{ \nabla} \cdot F) = 0\text{.}\)
5. True or false? If \(\vecs{F} \) is a vector field on \(\mathbb{R}^3\) such that \(|\vecs{F} (x, y, z)| = 1\) for all \(x\text{,}\) \(y\text{,}\) \(z\text{,}\) and if \(S\) is the sphere \(x^2 + y^2 + z^2 = 1\text{,}\) then \(\iint_S \vecs{F} \cdot\hat{\textbf{n}}\,\text{d}S = 4\pi\text{.}\)
6. True or false? Every closed surface \(S\) in \(\mathbb{R}^3\) is orientable. (Recall that \(S\) is closed if it is the boundary of a solid region \(E\text{.}\))

4 ✳

1. In the curve shown below (a helix lying in the surface of a cone), is the curvature increasing, decreasing, or constant as z increases?

   

2. Of the two functions shown below, one is a function \(f(x)\) and one is its curvature \(\kappa(x)\text{.}\) Which is which?

   

3. Let \(C\) be the curve of intersection of the cylinder \(x^2 + z^2 = 1\) and the saddle \(xz = y\text{.}\) Parametrise \(C\text{.}\) (Be sure to specify the domain of your parametrisation.)
4. Let \(H\) be the helical ramp (also known as a helicoid) which revolves around the \(z\)-axis in a clockwise direction viewed from above, beginning at the y-axis when \(z = 0\text{,}\) and rising \(2\pi\) units each time it makes a full revolution. Let \(S\) be the the portion of \(H\) which lies outside the cylinder \(x^2 + y^2 = 4\text{,}\) above the \(z = 0\) plane and below the \(z = 5\) plane. Choose one of the following functions and give the domain on which the function you have chosen parametrizes S. (Hint: Only one of the following functions is possible.)
   1. \(\displaystyle \vecs{r} (u, v) = \big(u \cos v, u \sin v, u\big)\)
   2. \(\displaystyle \vecs{r} (u, v) = \big(u \cos v, u \sin v, v\big)\)
   3. \(\displaystyle \vecs{r} (u, v) = \big(u \sin v, u \cos v, u\big)\)
   4. \(\displaystyle \vecs{r} (u, v) = \big(u \sin v, u \cos v, v\big)\)
5. Write down a parametrized curve of zero curvature and arclength \(1\text{.}\) (Be sure to specify the domain of your parametrisation.)
6. If \(\vecs{ \nabla} \cdot\vecs{F} \) is a constant \(C\) on all of \(\mathbb{R}^3\text{,}\) and \(S\) is a cube of unit volume such that the flux outward through each side of \(S\) is \(1\text{,}\) what is \(C\text{?}\)
7. Let

   \[ \vecs{F} (x, y) = \big(ax + by\,,\, cx + dy\big) \nonumber \]

   Give the full set of \(a\text{,}\) \(b\text{,}\) \(c\) and \(d\) such that \(\vecs{F} \) is conservative.
8. If \(\vecs{r} (s)\) has been parametrized by arclength (i.e. \(s\) is arclength), what is the arclength of \(\vecs{r} (s)\) between \(s = 3\) and \(s = 5\text{?}\)
9. Let \(\vecs{F} \) be a 2D vector field which is defined everywhere except at the points marked \(P\) and \(Q\text{.}\) Suppose that \(\vecs{ \nabla} \times\vecs{F} = 0\) everywhere on the domain of \(\vecs{F} \text{.}\) Consider the five curves \(R\text{,}\) \(S\text{,}\) \(T\text{,}\) \(U\text{,}\) and \(V\) shown in the picture.

   Which of the following is necessarily true?

   

   1. \(\displaystyle \int_S \vecs{F} \cdot\text{d}\vecs{r} = \int_T \vecs{F} \cdot\text{d}\vecs{r} \)
   2. \(\displaystyle \int_R \vecs{F} \cdot\text{d}\vecs{r} = \int_S \vecs{F} \cdot\text{d}\vecs{r} = \int_T \vecs{F} \cdot\text{d}\vecs{r} = \int_U \vecs{F} \cdot\text{d}\vecs{r} = 0\)
   3. \(\displaystyle \int_R \vecs{F} \cdot\text{d}\vecs{r} + \int_S \vecs{F} \cdot\text{d}\vecs{r} + \int_T \vecs{F} \cdot\text{d}\vecs{r} = \int_U \vecs{F} \cdot\text{d}\vecs{r} \)
   4. \(\displaystyle \int_U \vecs{F} \cdot\text{d}\vecs{r} = \int_R \vecs{F} \cdot\text{d}\vecs{r} + \int_S \vecs{F} \cdot\text{d}\vecs{r} \)
   5. \(\displaystyle \int_V \vecs{F} \cdot\text{d}\vecs{r} = 0\)
10. Write down a 3D vector field \(\vecs{F} \) such that for all closed surfaces \(S\text{,}\) the volume enclosed by \(S\) is equal to

    \[ \iint_S \vecs{F} \cdot \hat{\textbf{n}}\,\text{d}S \nonumber \]

11. Consider the vector field \(\vecs{F} \) in the \(xy\)-plane shown below. Is the \(\hat{\mathbf{k}}^{\rm th}\) component of \(\vecs{ \nabla} \times\vecs{F} \) at \(P\) positive, negative or zero?

5 ✳

Say whether the following statements are true or false.

1. If \(\vecs{F} \) is a 3D vector field defined on all of \(\mathbb{R}^3\), and \(S_1\) and \(S_2\) are two surfaces with the same boundary, but \(\iint_{S_1} \vecs{F} \cdot\hat{\textbf{n}}\,\text{d}S \ne \iint_{S_2} \vecs{F} \cdot\hat{\textbf{n}}\,\text{d}S\text{,}\) then \(\vecs{ \nabla} \cdot\vecs{F} \) is not zero anywhere.
2. If \(\vecs{F} \) is a vector field satisfying \(\vecs{ \nabla} \times\vecs{F} \) = 0 whose domain is not simply-connected, then \(\vecs{F} \) is not conservative.
3. The osculating circle of a curve \(C\) at a point has the same unit tangent vector, unit normal vector, and curvature as \(C\) at that point.
4. A planet orbiting a sun has period proportional to the cube of the major axis of the orbit.
5. For any 3D vector field \(\vecs{F} \text{,}\) \(\vecs{ \nabla} \cdot(\vecs{ \nabla} \times\vecs{F} )\) = 0.
6. A field whose divergence is zero everywhere in its domain has closed surfaces \(S\) in its domain.
7. The gravitational force field is conservative.
8. If \(\vecs{F} \) is a field defined on all of \(\mathbb{R}^3\) such that \(\int_C \vecs{F} \cdot \text{d}\vecs{r} = 3\) for some curve \(C\text{,}\) then \(\vecs{ \nabla} \times\vecs{F} \) is non-zero at some point.
9. The normal component of acceleration for a curve of constant curvature is constant.
10. The curve defined by

    \[ \vecs{r} _1(t) = \cos(t^4)\,\hat{\pmb{\imath}} + 3t^4\hat{\pmb{\jmath}},\qquad -\infty \lt t \lt \infty, \nonumber \]

    is the same as the curve defined by

    \[ \vecs{r} _2(t) = \cos t\,\hat{\pmb{\imath}} + 3t\,\hat{\mathbf{k}},\qquad -\infty \lt t \lt \infty \nonumber \]

6 ✳

Which of the following statements are true (T) and which are false (F)?

All real valued functions \(f(x,y,z)\) and all vector fields \(\vecs{F} (x, y, z)\) have domain \(\mathbb{R}^3\) unless specified otherwise.

1. If \(f\) is a continuous real valued function and \(S\) a smooth oriented surface, then

   \[ \iint_S f\, \text{d}S = -\iint_{-S} f\,\text{d}S \nonumber \]

   where `\(-S\)' denotes the surface \(S\) but with the opposite orientation.
2. Suppose the components of the vector field \(\vecs{F} \) have continuous partial derivatives. If \(\iint_S\vecs{ \nabla} \times\vecs{F} \cdot\hat{\textbf{n}}\,\text{d}S=0\) for every closed smooth surface, then \(\vecs{F} \) is conservative.
3. Suppose \(S\) is a smooth surface bounded by a smooth simple closed curve \(C\text{.}\) The orientation of \(C\) is determined by that of \(S\) as in Stokes' theorem. Suppose the real valued function \(f\) has continuous partial derivatives. Then

   \[ \int_C f\,\text{d}x =\iint_S \left(\frac{\partial f}{\partial z}\hat{\pmb{\jmath}} - \frac{\partial f}{\partial y}\hat{\mathbf{k}}\right)\cdot\hat{\textbf{n}}\,\text{d}S \nonumber \]

4. Suppose the real valued function \(f(x,y,z)\) has continuous second order partial derivatives. Then

   \[ (\vecs{ \nabla} f ) \times (\vecs{ \nabla} f ) = \vecs{ \nabla} \times (\vecs{ \nabla} f ) \nonumber \]

5. The curve parameterized by

   \[ \vecs{r} (t) = \big(2 + 4t^3 \,,\, -t^3 \,,\, 1 - 2t^3\big)\qquad -\infty \lt t \lt \infty \nonumber \]

   has curvature \(\kappa(t) = 0\) for all \(t\text{.}\)
6. If a smooth curve is parameterized by \(\vecs{r} (s)\) where \(s\) is arc length, then its tangent vector satisfies

   \[ |\vecs{r} '(s)| = 1 \nonumber \]

7. If \(S\) is the sphere \(x^2 + y^2 + z^2 = 1\) and \(\vecs{F} \) is a constant vector field, then \(\iint_S \vecs{F} \cdot\hat{\textbf{n}}\, \text{d}S = 0\text{.}\)
8. There exists a vector field \(\vecs{F} \) whose components have continuous second order partial derivatives such that \(\vecs{ \nabla} \times\vecs{F} = (x, y, z)\text{.}\)

7 ✳

The vector field \(\vecs{F} = P (x, y)\,\hat{\pmb{\imath}} + Q(x, y)\,\hat{\pmb{\jmath}}\) is plotted below.

In the following questions, give the answer that is best supported by the plot.

1. The derivative \(P_y\) at the point labelled \(A\) is (a) positive, (b) negative, (c) zero, (d) there is not enough information to tell.
2. The derivative \(Q_x\) at the point labelled \(A\) is (a) positive, (b) negative, (c) zero, (d) there is not enough information to tell.
3. The curl of \(\vecs{F} \) at the point labelled \(A\) is (a) in the direction of \(+\hat{\mathbf{k}}\) (b) in the direction of \(-\hat{\mathbf{k}}\) (c) zero (d) there is not enough information to tell.
4. The work done by the vector field on a particle travelling from point \(B\) to point \(C\) along the curve \(\mathcal{C}_1\) is (a) positive (b) negative (c) zero (d) there is not enough information to tell.
5. The work done by the vector field on a particle travelling from point \(B\) to point \(C\) along the curve \(\mathcal{C}_2\) is (a) positive (b) negative (c) zero (d) there is not enough information to tell.
6. The vector field \(\vecs{F} \) is (a) the gradient of some function \(f\) (b) the curl of some vector field \(\textbf{G}\) (c) not conservative (d) divergence free.

8 ✳

Which of the following statements are true (T) and which are false (F)?

1. The curve defined by

   \[ \vecs{r} _1(t) = \cos(t^2 )\,\hat{\pmb{\imath}} + \sin(t^2 )\,\hat{\pmb{\jmath}} + 2t^2\,\hat{\mathbf{k}},\qquad -\infty \lt t \lt \infty \nonumber \]

   is the same as the curve defined by

   \[ \vecs{r} _2 (t) = \cos t\,\hat{\pmb{\imath}} + \sin t\,\hat{\pmb{\jmath}} + 2t\,\hat{\mathbf{k}},\qquad -\infty \lt t \lt \infty \nonumber \]

2. The curve defined by

   \[ \vecs{r} _1(t) = \cos(t^2)\,\hat{\pmb{\imath}} + \sin(t^2 )\,\hat{\pmb{\jmath}} + 2t^2\,\hat{\mathbf{k}},\qquad 0 \le t \le 1 \nonumber \]

   is the same as the curve defined by

   \[ \vecs{r} _2(t) = \cos t\,\hat{\pmb{\imath}} + \sin t\,\hat{\pmb{\jmath}} + 2t\,\hat{\mathbf{k}},\qquad 0 \le t \le 1 \nonumber \]

3. If a smooth curve is parameterized by \(\vecs{r} (s)\) where \(s\) is arc length, then its tangent vector satisfies

   \[ |\vecs{r} '(s)| = 1 \nonumber \]

4. If \(\vecs{r} (t)\) defines a smooth curve \(C\) in space that has constant curvature \(\kappa \gt 0\text{,}\) then \(C\) is part of a circle with radius \(1/\kappa\text{.}\)
5. If the speed of a moving object is constant, then its acceleration is orthogonal to its velocity.
6. The vector field

   \[ \vecs{F} (x, y, z) = \frac{-y}{x^2+y^2} \hat{\pmb{\imath}} + \frac{x}{x^2+y^2} \hat{\pmb{\jmath}} + z\hat{\mathbf{k}} \nonumber \]

   is conservative.
7. Suppose the vector field \(\vecs{F} (x, y, z)\) is defined on an open domain and its components have continuous partial derivatives. If \(\vecs{ \nabla} \times\vecs{F} = 0\text{,}\) then \(\vecs{F} \) is conservative.
8. The region \(D =\left \{ (x, y) \big|B x^2 + y^2 \gt 1 \right \}\) is simply connected.
9. The region \(D = \left \{ (x, y) \big|B y - x^2 \gt 0 \right \}\) is simply connected.
10. If \(\vecs{F} \) is a vector field whose components have two continuous partial derivatives, then

    \[ \iint_S \vecs{ \nabla} \times \vecs{F} \cdot\hat{\textbf{n}}\, \text{d}S = 0 \nonumber \]

    when \(S\) is the boundary of a solid region \(E\) in \(\mathbb{R}^3\text{.}\)

9 ✳

Which of the following statements are true (T) and which are false (F)?

1. If a smooth curve \(C\) is parameterized by \(\vecs{r} (s)\) where \(s\) is arc length, then the tangent vector \(\vecs{r} '(s)\) satisfies \(|\vecs{r} '(s)| = 1\text{.}\)
2. If \(\vecs{r} (t)\) defines a smooth curve \(C\) in space that has constant curvature \(\kappa \gt 0\text{,}\) then \(C\) is part of a circle with radius \(1/\kappa\text{.}\)
3. Suppose \(\vecs{F} \) is a continuous vector field with open domain \(D\text{.}\) If

   \[ \int_C \vecs{F} \cdot \text{d}\vecs{r} = 0 \nonumber \]

   for every piecewise smooth closed curve \(C\) in \(D\text{,}\) then \(\vecs{F} \) is conservative.
4. Suppose \(\vecs{F} \) is a vector field with open domain \(D\text{,}\) and the components of \(\vecs{F} \) have continuous partial derivatives. If \(\vecs{ \nabla} \times \vecs{F} = 0\) everywhere on \(D\text{,}\) then \(\vecs{F} \) is conservative.
5. The curve defined by

   \[ \vecs{r} _1(t) = \cos(t^2)\,\hat{\pmb{\imath}} + \sin(t^2)\,\hat{\pmb{\jmath}} + 2t^2\,\hat{\mathbf{k}},\qquad -\infty \lt t \lt \infty \nonumber \]

   is the same as the curve defined by

   \[ \vecs{r} _2 (t) = \cos t\,\hat{\pmb{\imath}} + \sin t\,\hat{\pmb{\jmath}} + 2t\,\hat{\mathbf{k}},\qquad -\infty \lt t \lt \infty \nonumber \]

6. The curve defined by

   \[ \vecs{r} _1(t) = \cos(t^2)\,\hat{\pmb{\imath}} + \sin(t^2)\,\hat{\pmb{\jmath}} + 2t^2\,\hat{\mathbf{k}},\qquad 0\le t \le 1 \nonumber \]

   is the same as the curve defined by

   \[ \vecs{r} _2 (t) = \cos t\,\hat{\pmb{\imath}} + \sin t\,\hat{\pmb{\jmath}} + 2t\,\hat{\mathbf{k}},\qquad 0\le t\le 1 \nonumber \]

7. Suppose \(\vecs{F} (x, y, z)\) is a vector field whose components have continuous second order partial derivatives. Then \(\vecs{ \nabla} \cdot (\vecs{ \nabla} \times F) = 0\text{.}\)
8. Suppose the real valued function \(f(x, y, z)\) has continuous second order partial derivatives. Then \(\vecs{ \nabla} \cdot(\vecs{ \nabla} f) = 0\text{.}\)
9. The region \(D = \left \{ (x, y) \big| x^2 + y^2 \gt 1 \right \}\) is simply connected.
10. The region \(D = \left \{ (x, y) \big| y - x^2 \gt 0 \right \}\) is simply connected.

10 ✳

Let \(\vecs{F} \text{,}\) \(\textbf{G}\) be vector fields, and \(f\text{,}\) \(g\) be scalar fields. Assume \(\vecs{F} \text{,}\) \(\textbf{G}\text{,}\) \(f\text{,}\) \(g\) are defined on all of \(\mathbb{R}^3\) and have continuous partial derivatives of all orders everywhere. Mark each of the following as True (T) or False (F).

1. If \(C\) is a closed curve and \(\vecs{ \nabla} f=\vecs{0}\text{,}\) then \(\int_C f\,\text{d}s=0\text{.}\)
2. If \(\vecs{r} (t)\) is a parametrization of a smooth curve \(C\) and the binormal \(\textbf{B}(t)\) is constant then \(C\) is a straight line.
3. If \(\vecs{r} (t)\) is the position of a particle which travels with constant speed, then \(\vecs{r} '(t)\cdot\vecs{r} ''(t)=0\text{.}\)
4. If \(C\) is a path from points \(A\) to \(B\text{,}\) then the line integral \(\int_C\big(\vecs{F} \times\textbf{G}\big)\cdot\text{d}\vecs{r} \) is independent of the path \(C\text{.}\)
5. The line integral \(\int_C f\,\text{d}s\) does not depend of the orientation of the curve \(C\text{.}\)
6. If \(S\) is a parametric surface \(\vecs{r} (u,v)\) then a normal to \(S\) is given by

   \[\begin{gather*} \frac{\partial\vecs{r} }{\partial u}\times \frac{\partial\vecs{r} }{\partial u} \end{gather*}\]

7. The surface area of the parametric surface \(S\) given by \(\vecs{r} (u,v) = x(u,v)\,\hat{\pmb{\imath}} + y(u,v)\,\hat{\pmb{\jmath}} + z(u,v)\,\hat{\mathbf{k}}\text{,}\) \((u,v)\in D\text{,}\) is given by

   \[\begin{gather*} \iint_D \left(1+\big(\tfrac{\partial z}{\partial u}\big)^2 +\big(\tfrac{\partial z}{\partial v}\big)^2\right)^{1/2} \text{d}u\text{d}v \end{gather*}\]

8. If \(\vecs{F} \) is the velocity field of an incompressible fluid then \(\vecs{ \nabla} \cdot\vecs{F} =0\text{.}\)
9. \(\displaystyle \vecs{ \nabla} \cdot\big(\vecs{F} \times\textbf{G}\big) = (\vecs{ \nabla} \cdot\vecs{F} )\textbf{G} + (\vecs{ \nabla} \cdot\textbf{G})\vecs{F} \)

11 ✳

Say whether the following statements are true (T) or false (F). You may assume that all functions and vector fields are defined everywhere and have derivatives of all orders everywhere.

1. The divergence of \(\vecs{ \nabla} \times\vecs{F} \) is zero, for every \(\vecs{F} \text{.}\)
2. In a simply connected region, \(\int_C \vecs{F} \cdot\text{d}\vecs{r} \) depends only on the endpoints of \(C\text{.}\)
3. If \(\vecs{ \nabla} f = 0\text{,}\) then \(f\) is a constant function.
4. If \(\vecs{ \nabla} \times\vecs{F} = \vecs{0}\text{,}\) then \(\vecs{F} \) is a constant vector field.
5. If \(\vecs{ \nabla} \cdot\vecs{F} = 0\text{,}\) then \(\iint_S\vecs{F} \cdot\hat{\textbf{n}}\, \text{d}S = 0\) for every closed surface \(S\text{.}\)
6. If \(\int_C \vecs{F} \cdot \text{d}\vecs{r} = 0\) for every closed curve \(C\text{,}\) then \(\vecs{ \nabla} \times\vecs{F} = 0\text{.}\)
7. If \(\vecs{r} (t)\) is a path in three space with constant speed \(|\vecs{v} (t)|\text{,}\) then the acceleration is perpendicular to the tangent vector, i.e. \(\textbf{a}\cdot\hat{\textbf{T}} = 0\text{.}\)
8. If \(\vecs{r} (t)\) is a path in three space with constant curvature \(\kappa\text{,}\) then \(\vecs{r} (t)\) parameterizes part of a circle of radius \(1/\kappa\text{.}\)
9. Let \(\vecs{F} \) be a vector field and suppose that \(S_1\) and \(S_2\) are oriented surfaces with the same boundary curve \(C\text{,}\) and \(C\) is given the direction that is compatible with the orientations of \(S_1\) and \(S_2\). Then \(\iint_{S 1} \vecs{F} \cdot\hat{\textbf{n}}\, \text{d}S = \iint_{S 2} \vecs{F} \cdot\hat{\textbf{n}}\, \text{d}S\text{.}\)
10. Let \(A(t)\) be the area swept out by the trajectory of a planet from time \(t=0\) to time \(t\text{.}\) The \(\dfrac{dA}{dt}\) is constant.

12 ✳

Find the correct identity, if \(f\) is a function and \(\textbf{G}\) and \(\vecs{F} \) are vector fields. Select the true statement.

1. \(\displaystyle \vecs{ \nabla} \cdot(f \vecs{F} ) = f \vecs{ \nabla} \times(\vecs{F} ) + (\vecs{ \nabla} f ) \times\vecs{F} \)
2. \(\displaystyle \vecs{ \nabla} \cdot(f \vecs{F} ) = f \vecs{ \nabla} \cdot(\vecs{F} ) + \vecs{F} \cdot \vecs{ \nabla} f\)
3. \(\displaystyle \vecs{ \nabla} \times(f \vecs{F} ) = f \vecs{ \nabla} \cdot(\vecs{F} ) + \vecs{F} \cdot \vecs{ \nabla} f\)
4. None of the above are true.

13 ✳

True or False. Consider vector fields \(\vecs{F} \) and scalar functions \(f\) and \(g\) which are defined and smooth in all of three-dimensional space. Let \(\vecs{r} =(x,y,z)\) represent a variable point in space, and let \(\boldsymbol{\omega} = (\omega_1,\omega_2,\omega_3)\) be a constant vector. Let \(\Omega\) be a smoothly bounded domain with outer normal \(\hat{\textbf{n}}\text{.}\) Which of the following are identites, always valid under these assumptions?

1. \(\displaystyle \vecs{ \nabla} \cdot\vecs{ \nabla} f = 0\)
2. \(\displaystyle \vecs{F} \times\vecs{ \nabla} f = f\,\vecs{ \nabla} \cdot\vecs{F} \)
3. \(\displaystyle \vecs{ \nabla} ^2 f = \vecs{ \nabla} (\vecs{ \nabla} \cdot f)\)
4. \(\displaystyle \vecs{ \nabla} \times\vecs{ \nabla} f = \vecs{0}\)
5. \(\displaystyle (\vecs{ \nabla} \times f)+(\vecs{ \nabla} \times g) = \vecs{ \nabla} f\times\vecs{ \nabla} g\)
6. \(\displaystyle \vecs{ \nabla} \cdot\vecs{ \nabla} \times\vecs{F} = 0\)
7. \(\vecs{ \nabla} \cdot\frac{\vecs{r} }{|\vecs{r} |^2} = 0\) for \(\vecs{r} \ne\vecs{0}\)
8. \(\displaystyle \vecs{ \nabla} \times(\boldsymbol{\omega}\times\vecs{r} ) = \vecs{0}\)
9. \(\displaystyle \iiint_\Omega f\vecs{ \nabla} \cdot\vecs{F} \,\text{d}V =-\iiint_\Omega \vecs{ \nabla} f\cdot\vecs{F} \,\text{d}V +\iint_{\partial\Omega} f\vecs{F} \cdot\hat{\textbf{n}}\,\text{d}S\)
10. \(\displaystyle \iint_{\partial\Omega} f\hat{\textbf{n}}\,\text{d}S =- \iiint_\Omega \vecs{ \nabla} f\,\text{d}V\)

14 ✳

Determine if the given statements are True or False. Provide a reason or a counterexample.

1. A constant vector field is conservative on \(\mathbb{R}^3\text{.}\)
2. If \(\vecs{ \nabla} \cdot\vecs{F} = 0\) for all points in the domain of \(\vecs{F} \) then \(\vecs{F} \) is a constant vector field.
3. Let \(\vecs{r} (t)\) be a parametrization of a curve \(C\) in \(\mathbb{R}^3\text{.}\) If \(\vecs{r} (t)\) and \(\dfrac{d\vecs{r} }{dt}\) are orthogonal at all points of the curve \(C\text{,}\) then \(C\) lies on the surface of a sphere \(x^2 + y^2 + z^2 = a^2\) for some \(a \gt 0\text{.}\)
4. The curvature \(\kappa\) at a point on a curve depends on the orientation of the curve.
5. The domain of a conservative vector field must be simply connected.

15 ✳

Provide a short answer to each question.

1. Compute \(\vecs{ \nabla} \cdot\big(x^2 y\,\hat{\pmb{\imath}} + e^y \sin x\,\hat{\pmb{\jmath}} + e^{zx}\,\hat{\mathbf{k}}\big)\)
2. Compute \(\vecs{ \nabla} \times(\cos x^2\,\hat{\pmb{\imath}} - y^3 z\,\hat{\pmb{\jmath}} + xz\,\hat{\mathbf{k}}\big)\)
3. Let

   \[ \vecs{F} = \frac{x}{x^2+y^2}\hat{\pmb{\imath}} +\frac{y}{x^2+y^2}\hat{\pmb{\jmath}} +z^2\,\hat{\mathbf{k}} \nonumber \]

   and let \(D\) be the domain of \(\vecs{F} \text{.}\) Consider the following four statments.

   1. \(D\) is connected
   2. \(D\) is disconnected
   3. \(D\) is simply connected
   4. \(D\) is not simply connected

   Choose one of the following:

   1. (II) and (III) are true
   2. (I) and (III) are true
   3. (I) and (IV) are true
   4. (II) and (IV) are true
   5. Not enough information to determine
4. True or False? If the speed of a particle is constant then the acceleration of the particle is zero. If your answer is True, provide a reason. If your answer is False, provide a counter example.

16 ✳

Are each of the following statements True or False? Recall that \(f \in C^k\) means that all derivatives of \(f\) up to order \(k\) exist and are continuous.

1. \(\vecs{ \nabla} \times(f \vecs{ \nabla} f ) = \vecs{0}\) for all \(C^2\) scalar functions \(f\) in \(\mathbb{R}^3\text{.}\)
2. \(\vecs{ \nabla} \cdot(f\vecs{F} ) = \vecs{ \nabla} f \cdot\vecs{F} + f\vecs{ \nabla} \cdot \vecs{F} \) for all \(C^1\) scalar functions \(f\) and \(C^1\) vector fields \(\vecs{F} \) in \(\mathbb{R}^3\text{.}\)
3. A smooth space curve \(C\) with constant curvature \(\kappa = 0\) must be a part of a straight line.
4. A smooth space curve \(C\) with constant curvature \(\kappa \ne 0\) must be part of a circle of radius \(1/\kappa\text{.}\)
5. If \(f\) is any smooth function defined in \(\mathbb{R}^3\) and if \(C\) is any circle, then \(\int_C\vecs{ \nabla} f\cdot\text{d}\vecs{r} =0\text{.}\)
6. Suppose \(\vecs{F} \) is a smooth vector field in \(\mathbb{R}^3\) and \(\vecs{ \nabla} \cdot\vecs{F} =0\) everywhere. Then, for every sphere, the flux out of one hemisphere is equal to the flux into the opposite hemisphere.
7. Let \(\vecs{F} (x, y,z)\) be a continuously differentiable vector field which is defined for every \((x, y, z)\text{.}\) Then, \(\iint_S\vecs{ \nabla} \times\vecs{F} \cdot\hat{\textbf{n}}\,\text{d}S=0\) for any closed surface \(S\text{.}\) (A closed surface is a surface that is the boundary of a solid region.)

17 ✳

True or false (reasons must be given):

1. If a smooth vector field on \(\mathbb{R}^3\) is curl free and divergence free, then its potential is harmonic. By definition, \(\phi(x,y,z)\) is harmonic if \(\big(\frac{\partial^2 }{\partial x^2} +\frac{\partial^2 }{\partial y^2} +\frac{\partial^2 }{\partial z^2}\big) \phi(x,y,z)=0\text{.}\)
2. If \(\vecs{F} \) is a smooth conservative vector field on \(\mathbb{R}^3\text{,}\) then its flux through any smooth closed surface is zero.

18 ✳

The following statements may be true or false. Decide which. If true, give a proof. If false, provide a counter-example.

1. If \(f\) is any smooth function defined in \(\mathbb{R}^3\) and if \(C\) is any circle, then \(\int_{C}\vecs{ \nabla} f \cdot \text{d}\vecs{r} =0\text{.}\)
2. There is a vector field \(\vecs{F} \) that obeys \(\vecs{ \nabla} \times\vecs{F} =x\,\hat{\pmb{\imath}}+y\,\hat{\pmb{\jmath}}+z\,\hat{\mathbf{k}}\text{.}\)

19 ✳

Short answers:

1. Let \(S\) be the level surface \(f(x,y,z)=0\text{.}\) Why is \(\int_C \vecs{ \nabla} f\cdot \text{d}\vecs{r} =0\) for any curve \(C\) on \(S\text{?}\)
2. A point moving in space with position \(\vecs{r} (t)\) at time \(t\) satisfies the condition \(\textbf{a}(t)=f(t)\vecs{r} (t)\) for all \(t\) for some real valued function \(f\text{.}\) Why is \(\vecs{v} \times\vecs{r} \) a constant vector?
3. Why is the trajectory of the point in (b) contained in a plane?
4. Is the binormal vector, \(\hat{\textbf{B}}\text{,}\) of a particle moving in space, always orthogonal to the unit tangent vector \(\hat{\textbf{T}}\) and unit normal \(\hat{\textbf{N}}\text{?}\)
5. If the curvature of the path of a particle moving in space is constant, is the acceleration zero when maximum speed occurs?

20 ✳

A region \(R\) is bounded by a simple closed curve \(\mathcal{C}\text{.}\) The curve \(\mathcal{C}\) is oriented such that \(R\) lies to the left of \(\mathcal{C}\) when walking along \(\mathcal{C}\) in the direction of \(\mathcal{C}\text{.}\) Determine whether or not each of the following expressions is equal to the area of \(R\text{.}\) You must justify your conclusions.

1. \(\displaystyle \frac{1}{2} \int_\mathcal{C} -y \,d x +x \,d y\)
2. \(\displaystyle \frac{1}{2} \int_\mathcal{C} -x \,d x + y \,d y\)
3. \(\displaystyle \int_\mathcal{C} y \,d x\)
4. \(\displaystyle \int_\mathcal{C} 3y\,d x + 4x \,d y\)

21 ✳

Say whether each of the following statements is true or false and explain why.

1. A moving particle has velocity and acceleration vectors that satisfy \(|\vecs{v} | = 1\) and \(|\textbf{a}|=1\) at all times. Then the curvature of this particle's path is a constant.
2. If \(\vecs{F} \) is any smooth vector field defined in \(\mathbb{R}^3\) and if \(S\) is any sphere, then

   \[ \iint_{S}\vecs{ \nabla} \times\vecs{F} \cdot\hat{\textbf{n}}\,\text{d}S=0 \nonumber \]

   Here \(\hat{\textbf{n}}\) is the outward normal to \(S\text{.}\)
3. If \(\vecs{F} \) and \(\textbf{G}\) are smooth vector fields in \(\mathbb{R}^3\) and if \(\displaystyle \oint_C \vecs{F} \cdot \text{d}\vecs{r} =\oint_C \textbf{G}\cdot \text{d}\vecs{r} \) for every circle \(C\text{,}\) then \(\vecs{F} =\textbf{G}\text{.}\)

22 ✳

Three quickies:

1. A moving particle with position \(\vecs{r} (t) = (x(t),y(t),z(t))\) satisfies

   \[ \textbf{a} = f(\vecs{r} ,\vecs{v} )\vecs{r} \nonumber \]

   for some scalar-valued function \(f\text{.}\) Prove that \(\vecs{r} \times\vecs{v} \) is constant.
2. Calculate \(\iint_\mathcal{S}(x\,\hat{\pmb{\imath}} - y\,\hat{\pmb{\jmath}} + z^2\,\hat{\mathbf{k}})\cdot \hat{\textbf{n}}\text{d}S\text{,}\) where \(\mathcal{S}\) is the boundary of any solid right circular cylinder of radius \(b\) with one base in the plane \(z=1\) and the other base in the plane \(z=3\text{.}\)
3. Let \(\vecs{F} \) and \(\textbf{G}\) be smooth vector fields defined in \(\mathbb{R}^3\text{.}\) Suppose that, for every circle \(C\text{,}\) we have \(\oint_{C} \vecs{F} \cdot \text{d}\vecs{r} =\iint_S \textbf{G}\cdot \hat{\textbf{n}}\,\text{d}S\text{,}\) where \(S\) is the oriented disk with boundary \(C\text{.}\) Prove that \(\textbf{G}=\vecs{ \nabla} \times\vecs{F} \text{.}\)