Skip to main content
\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)
Mathematics LibreTexts

16.E: Vector Calculus (Exercises)

16.1: Vector Fields

16.5: Divergence and Curl

 

For the following exercises, determine whether the statement is true or false.

If the coordinate functions of \(F : \mathbb{R}^3 \rightarrow \mathbb{R}^3\) have continuous second partial derivatives, then \(curl \, (div (F))\) equals zero.

\(\nabla \cdot (xi + yj + zk) = 1\).

[Hide Solution]

False

All vector fields of the form

\(F(x,y,z) = f(x)i + g(y)j + h(z)k\) are conservative.

If \(curl \, F = 0\), then F is conservative.

[Hide Solution]

True

If F is a constant vector field then \(div \, F = 0\).

If F is a constant vector field then \(curl \, F = 0\).

[Hide Solution]

True

 

For the following exercises, find the curl of F.

\(F(x,y,z) = xy^2z^4 i + (2x^2y + z)j + y^3 z^2 k\)

\(F(x,y,z) = x^2 zi + y^2 xj + (y + 2z) k\)

[Hide Solution]

\(curl \, F = i + x^2 j + y^2 k\)

 

\(F(x,y,z) = 3xyz^2i + y^2 \sin z j + xe^{2z} k\)

\(F(x,y,z) = x^2 yzi + xy^2 zj + xyz^2 k\)

[Hide Solution]

\(curl \, F = (xz^2 - xy^2)i + (x^2 y - yz^2)j + (y^2z - x^2z) k\)

 

\(F(x,y,z) = (x \, \cos y)i + xy^2j\)

\(F(x,y,z) = (x - y)i + (y - z)j + (z - x) k\)

[Hide Solution]

\(curl \, F = i + j + k\)

 

\(F(x,y,z) = xyzi + x^2y^2z^2j + y^2z^3k\)

\(F(x,y,z) = xyi + yzj + xz k\)

[Hide Solution]

\(curl \, F = - yi - zj - xk\)

 

\(F(x,y,z) = x^2 i + y^2 j + z^2 k\)

\(F(x,y,z) = axi + byj + c k\) for constants a, b, c

[Hide Solution]

\(curl \, F = 0\)

 

For the following exercises, find the divergence of F.

\(F(x,y,z) = x^2 z i + y^2 x j + (y + 2z) k\)

\(F(x,y,z) = 3xyz^2 i + y^2 \sin z j + xe^2 k\)

[Hide Solution]

\(div F = 3yz^2 + 2y \, \sin z + 2xe^{2z}\)

 

\(F(x,y) = (\sin x) i + (\cos y) j\)

\(F(x,y,z) = x^2 i + y^2 j + z^2 k\)

[Hide Solution]

\(div \, F = 2(x + y + z)\)

 

\(F(x,y,z) = (x - y) i + (y - z) j + (z - x) k\)

\(F(x,y) = \frac{x}{\sqrt{x^2+y^2}}i + \frac{y}{\sqrt{x^2+y^2}}j\)

[Hide Solution]

\(div \, F = \frac{1}{\sqrt{x^2+y^2}}\)

 

\(F(x,y) = xi - yj\)

\(F(x,y,z) = axi + byj + ck\) for constants a, b, c

[ Hide Solution]

\(div \, F = a + b\)

 

\(F(x,y,z) = xyzi + x^2y^2z^2j + y^2z^3k\)

\(F(x,y,z) = xyi + yzj + xzk\)

[Hide Solution]

\(div \, F = x + y + z\)

 

For the following exercises, determine whether each of the given scalar functions is harmonic.

\(u(x,y,z) = e^{-x} (\cos y - \sin y)\)

\(w(x,y,z) = (x^2 + y^2 + z^2)^{-1/2}\)

[Hide Solution]

Harmonic

 

If \(F(x,y,z) = 2 i + 2x j + 3y k\) and

\(G(x,y,z) = x i - y j + z k\), find \(curl \, (F \times G)\).

If \(F(x,y,z) = 2 i + 2x j + 3y k\) and

\(G(x,y,z) = x i - y j + z k\), find \(div \, (F \times G)\).

[Hide Solution]

\(div \, (F \times G) = 2z + 3x\)

 

Find \(div \, F\), given that \(F = \nabla f\), where \(f(x,y,z) = xy^3z^2\).

Find the divergence of F for vector field

\(F(x,y,z) = (y^2 + z^2) (x + y) i + (z^2 + x^2)(y + z) j + (x^2 + y^2)(z + x) k\).

[Hide Solution]

\(div \, F = 2r^2\)

 

Find the divergence of F for vector field

\(F(x,y,z) = f_1(y,z) i + f_2 (x,z) j + f_3 (x,y) k\).

 

For the following exercises, use \(r = |r|\) and \(r = (x,y,z)\).

Find the \(curl \, r\)

[Hide Solution]

\(curl \, r = 0\)

 

Find the \(curl \, \frac{r}{r}\).

Find the \(curl \, \frac{r}{r^3}\).

[Hide Solution]

\(curl \, \frac{r}{r^3} = 0\)

 

Let \(F(x,y) = \frac{-yi+xj}{x^2+y^2}\), where F is defined on \(\{(x,y) \in \mathbb{R} | (x,y) \neq (0,0) \}\). Find \(curl \, F\).

 

For the following exercises, use a computer algebra system to find the curl of the given vector fields.

[T] \(F(x,y,z) = arctan \left(\frac{x}{y}\right) i + \ln \sqrt{x^2 + y^2} j + k\)

[Hide Solution]

\(curl \, F = \frac{2x}{x^2+y^2} k\)

[T]

\(F(x,y,z) = \sin (x - y) i + \sin (y - z) j + \sin (z - x) k\)

 

For the following exercises, find the divergence of F at the given point.

\(F(x,y,z) = i + j + k\) at \((2, -1, 3)\)

[Hide Solution]

\(div \, F = 0\)

 

\(F(x,y,z) = xyz i + y j + zk\) at \((1, 2, 3)\)

\(F(x,y,z) = e^{-xy}i + e^{xz}j + e^{yz}k\) at \((3, 2, 0)\)

[Hide Solution]

\(div \, F = 2 - 2e^{-6}\)

 

\(F(x,y,z) = xyz i + y j + zk\) at \((1, 2, 1)\)

\(F(x,y,z) = e^x \sin y i - e^x \cos y j \) at \((0, 0, 3)\)

\(div \, F = 0\)

 

For the following exercises, find the curl of F at the given point.

\(F(x,y,z) = i + j + k\) at \((2, -1, 3)\)

\(F(x,y,z) = xyz i + y j + zk\) at \((1, 2, 3)\)

[Hide Solution]

\(curl \, F = j - 3k\)

 

\(F(x,y,z) = e^{-xy}i + e^{xz}j + e^{yz}k\) at \((3, 2, 0)\)

\(F(x,y,z) = xyz i + y j + zk\) at \((1, 2, 1)\)

[Hide Solution]

\(curl \, F = 2j - k\)

 

\(F(x,y,z) = e^x \sin y i - e^x \cos y j \) at \((0, 0, 3)\)

Let \(F(x,y,z) = (3x^2 y + az) i + x^3 j + (3x + 3z^2) k\).

For what value of a is F conservative?

[Hide Solution]

\(a = 3\)

 

Given vector field \(F(x,y) = \frac{1}{x^2+y^2} (-y,x)\) on domain \(D = \frac{\mathbb{R}^2}{\{(0,0)\}} = \{(x,y) \in \mathbb{R}^2 |(x,y) \neq (0,0) \}\), is F conservative?

Given vector field \(F(x,y) = \frac{1}{x^2+y^2} (x,y)\) on domain \(D = \frac{\mathbb{R}^2}{\{(0,0)\}}\), is F conservative?

[Hide Solution]

F is conservative.

 

Find the work done by force field \(F(x,y) = e^{-y}i - xe^{-y}j\) in moving an object from P(0, 1) to Q(2, 0). Is the force field conservative?

 

Compute divergence \(F = (\sinh x) i + (\cosh y) j - xyz k\).

[Hide Solution]

\(div \, F = \cosh x + \sinh y - xy\)

 

Compute \(curl \, F = (\sinh x) i + (\cosh y) j - xyz k\).

 

For the following exercises, consider a rigid body that is rotating about the x-axis counterclockwise with constant angular velocity \(\omega = \langle a,b,c \rangle\). If P is a point in the body located at \(r = xi + yj + zk\), the velocity at P is given by vector field \(F = \omega \times r\).

A three dimensional diagram of an object rotating about the x axis in a counterclockwise manner with constant angular velocity w = <a,b,c>. The object is roughly a sphere with pointed ends on the x axis, which cuts it in half. An arrow r is drawn from (0,0,0) to P(x,y,z) and down from P(x,y,z) to the x axis.

Express F in terms of i, j, and k vectors.

[Hide Solution]

\((bz - cy)i(cx - az)j + (ay - bx)k\)

 

Find \(div \, F\).

Find \(curl \, F\)

[Hide Solution]

\(curl \, F = 2\omega\)

 

In the following exercises, suppose that \(\nabla \cdot F = 0\) and \(\nabla \cdot G = 0\).

Does \(F + G\) necessarily have zero divergence?

Does \(F \times G\) necessarily have zero divergence?

[Hide Solution]

\(F \times G\) does not have zero divergence.

 

In the following exercises, suppose a solid object in \(\mathbb{R}^3\) has a temperature distribution given by \(T(x,y,z)\). The heat flow vector field in the object is \(F = - k \nabla T\), where \(k > 0\) is a property of the material. The heat flow vector points in the direction opposite to that of the gradient, which is the direction of greatest temperature decrease. The divergence of the heat flow vector is \(\nabla \cdot F = -k \nabla \cdot \nabla T = - k \nabla^2 T\).

Compute the heat flow vector field.

Compute the divergence.

[Hide Solution]

\(\nabla \cdot F = -200 k [1 + 2(x^2 + y^2 + z^2)] e^{-x^2+y^2+z^2}\)

 

[T] Consider rotational velocity field \(v = \langle 0,10z, -10y \rangle\). If a paddlewheel is placed in plane \(x + y + z = 1\) with its axis normal to this plane, using a computer algebra system, calculate how fast the paddlewheel spins in revolutions per unit time.

A three dimensional diagram of a rotational velocity field. The arrows are showing a rotation in a clockwise manner. A paddlewheel is shown in plan x + y + z = 1 with n extended out perpendicular to the plane.

Glossary

curl

the curl of vector field \(F = \langle P,Q,R \rangle\), denoted \(\nabla \times F\), is the “determinant” of the matrix \(\begin{vmatrix} i & j & k \nonumber \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \nonumber \\ P & Q & R \end{vmatrix}\) and is given by the expression \((R_y - Q_z)i + (P_z - R_x ) j + (Q_x - P_y) k\); it measures the tendency of particles at a point to rotate about the axis that points in the direction of the curl at the point

divergence

the divergence of a vector field \(F = \langle P,Q,R\rangle \), denoted \(\nabla \times F\), is \(P_x + Q_y + R_z\); it measures the “outflowing-ness” of a vector field

16.7: Stokes’ Theorem

 

For the following exercises, without using Stokes’ theorem, calculate directly both the flux of \(curl \, F \cdot N\) over the given surface and the circulation integral around its boundary, assuming all are oriented clockwise.

\(F(x,y,z) = y^2i + z^2j + x^2k\); S is the first-octant portion of plane \(x + y + z = 1\).

\(F(x,y,z) = zi + xj + yk\); S is hemisphere \(z = (a^2 - x^2 - y^2)^{1/2}\).

[Hide Solution]

\[\iint_S (curl \, F \cdot N) \, dS = \pi a^2\]

\(F(x,y,z) = y^2i + 2xj + 5k\); S is hemisphere \(z = (4 - x^2 - y^2)^{1/2}\).

\(F(x,y,z) = zi + 2xj + 3yk\); S is upper hemisphere \(z = \sqrt{9 - x^2 - y^2}\).

[Hide Solution]

\[\iint_S (curl \, F \cdot N) \, dS = 18 \pi\]

 

\(F(x,y,z) = (x + 2z)i + (y - x)j + (z - y)k\); S is a triangular region with vertices (3, 0, 0), (0, 3/2, 0), and (0, 0, 3).

\(F(x,y,z) = 2yi + 6zj + 3xk\); S is a portion of paraboloid \(z = 4 - x^2 - y^2\) and is above the xy-plane.

[Hide Solution]

\[\iint_S (curl \, F \cdot N) \, dS = -8 \pi\]

 

For the following exercises, use Stokes’ theorem to evaluate

\[\iint_S (curl \, F \cdot N) \, dS\] for the vector fields and surface.

\(F(x,y,z) = xyi - zj\) and S is the surface of the cube \(0 \leq x \leq 1, \, 0 \leq y \leq 1, \, 0 \leq z \leq 1\), except for the face where \(z = 0\) and using the outward unit normal vector.

\(F(x,y,z) = xyi + x^2 j + z^2 k\); and C is the intersection of paraboloid \(z = x^2 + y^2\) and plane \(z = y\), and using the outward normal vector.

\[\iint_S (curl \, F \cdot N) \, dS = 0\]

 

\(F(x,y,z) = 4yi + z j + 2y k\); and C is the intersection of sphere \(x^2 + y^2 + z^2 = 4\) with plane \(z = 0\), and using the outward normal vector.

 

Use Stokes’ theorem to evaluate \[\int_C [2xy^2z \, dx + 2x^2yz \, dy + (x^2y^2 - 2z) \, dz],\] where C is the curve given by \(x = \cos t, \, y = \sin t, \, 0 \leq t \leq 2\pi\), traversed in the direction of increasing t.

A vector field in three dimensional space. The arrows are larger the further they are from the x, y plane. The arrows curve up from below the x, y plane and slightly above it. The rest tend to curve down and horizontally. An oval-shaped curve is drawn through the middle of the space.

[Hide Solution]

\[\int_C F \cdot dS = 0\]

[T] Use a computer algebraic system (CAS) and Stokes’ theorem to approximate line integral \[\int_C (y \, dx + z \, dy + x \, dz),\] where C is the intersection of plane \(x + y = 2\) and surface \(x^2 + y^2 + z^2 = 2(x + y)\), traversed counterclockwise viewed from the origin.

[T] Use a CAS and Stokes’ theorem to approximate line integral \[\int_C (3y\, dx + 2z \, dy - 5x \, dz),\] where C is the intersection of the xy-plane and hemisphere \(z = \sqrt{1 - x^2 - y^2}\), traversed counterclockwise viewed from the top—that is, from the positive z-axis toward the xy-plane.

[Hide Solution]

\[\int_C F \cdot dS = - 9.4248\]

[T] Use a CAS and Stokes’ theorem to approximate line integral \[ \int_C [(1 + y) \, z dx + (1 + z) x dy + (1 + x) y dz],\] where C is a triangle with vertices \((1,0,0), \, (0,1,0)\), and \((0,0,1)\) oriented counterclockwise.

 

Use Stokes’ theorem to evaluate \[\iint_S curl \, F \cdot dS,\] where \(F(x,y,z) = e^{xy} cos \, zi + x^2 zj + xyk\), and S is half of sphere \(x = \sqrt{1 - y^2 - z^2}\), oriented out toward the positive x-axis.

[Hide Solution]

\[\iint_S F \cdot dS = 0\]

 

[T] Use a CAS and Stokes’ theorem to evaluate \[\iint_S (curl \, F \cdot N) \, dS,\] where \(F(x,y,z) = x^2 yi + xy^2 j + z^3 k\) and C is the curve of the intersection of plane \(3x + 2y + z = 6\) and cylinder \(x^2 + y^2 = 4\), oriented clockwise when viewed from above.

[T] Use a CAS and Stokes’ theorem to evaluate \[\iint_S curl \, F \cdot dS,\] where \(F(x,y,z) = \left( \sin(y + z) - yx^2 - \frac{y^3}{3}\right)i + x \, \cos (y + z) j + \cos (2y) \, k\) and S consists of the top and the four sides but not the bottom of the cube with vertices \((\pm 1, \, \pm1, \, \pm1)\), oriented outward.

[Hide Solution]

\[\iint_S curl \, F \cdot dS = 2.6667\]

[T] Use a CAS and Stokes’ theorem to evaluate \[\iint_S curl \, F \cdot dS,\] where \(F(x,y,z) = z^2i + 3xyj + x^3y^3k\) and S is the top part of \(z = 5 - x^2 - y^2\) above plane \(z = 1\) and S is oriented upward.

 

Use Stokes’ theorem to evaluate \[\iint_S (curl \, F \cdot N) dS,\] where \(F(x,y,z) = z^2i + y^2j + xk\) and S is a triangle with vertices (1, 0, 0), (0, 1, 0) and (0, 0, 1) with counterclockwise orientation.

[Hide Solution]

\[\iint_S (curl \, F \cdot N)dS = -\frac{1}{6}\]

 

Use Stokes’ theorem to evaluate line integral \[\int_C (z \, dx + x \, dy + y \, dz),\] where C is a triangle with vertices (3, 0, 0), (0, 0, 2), and (0, 6, 0) traversed in the given order.

 

Use Stokes’ theorem to evaluate \[\int_C \left(\frac{1}{2} y^2 \, dx + z \, dy + x \, dz \right),\] where C is the curve of intersection of plane \(x + z = 1\) and ellipsoid \(x^2 + 2y^2 + z^2 = 1\), oriented clockwise from the origin.

A diagram of an intersecting plane and ellipsoid in three dimensional space. There is an orange curve drawn to show the intersection.

[Hide Solution]

\[\int_C \left(\frac{1}{2} y^2 \, dx + z \, dy + x \, dz \right) = - \frac{\pi}{4}\]

 

Use Stokes’ theorem to evaluate \[\iint_S (curl \, F \cdot N) dS,\] where \(F(x,y,z) = xi + y^2j + ze^{xy}k\) and S is the part of surface \(z = 1 - x^2 - 2y^2\) with \(z \geq 0\), oriented counterclockwise.

 

Use Stokes’ theorem for vector field \(F(x,y,z) = zi + 3xj + 2zk\) where S is surface \(z = 1 - x^2 - 2y^2, \, z \geq 0\), C is boundary circle \(x^2 + y^2 = 1\), and S is oriented in the positive z-direction.

[Hide Solution]

\[\iint_S (curl \, F \cdot N)dS = -3\pi]

 

Use Stokes’ theorem for vector field \(F(x,y,z) = - \frac{3}{2} y^2 i - 2 xyj + yzk\), where S is that part of the surface of plane \(x + y + z = 1\) contained within triangle C with vertices (1, 0, 0), (0, 1, 0), and (0, 0, 1), traversed counterclockwise as viewed from above.

 

A certain closed path C in plane \(2x + 2y + z = 1\) is known to project onto unit circle \(x^2 + y^2 = 1\)  in the xy-plane. Let c be a constant and let \(R(x,y,z) = xi + yj + zk\). Use Stokes’ theorem to evaluate \[\int_C(ck \times R) \cdot dS.\]

[Hide Solution]

\[\int_C (ck \times R) \cdot dS = 2\pi c\]

Use Stokes’ theorem and let C be the boundary of surface \(z = x^2 + y^2\) with \(0 \leq x \leq 2\) and \(0 \leq y \leq 1\) oriented with upward facing normal. Define

\(F(x,y,z) = [\sin (x^3) + xz] i + (x - yz)j + \cos (z^4) k\) and evaluate \(\int_C F \cdot dS\).

 

Let S be hemisphere \(x^2 + y^2 + z^2 = 4\) with \(z \geq 0\), oriented upward. Let \(F(x,y,z) = x^2 e^{yz}i + y^2 e^{xz} j + z^2 e^{xy}k\) be a vector field. Use Stokes’ theorem to evaluate \[\iint_S curl \, F \cdot dS.\]

[Hide Solution]

\[\iint_S curl \, F \cdot dS = 0\]

 

Let \(F(x,y,z) = xyi + (e^{z^2} + y)j + (x + y)k\) and let S be the graph of function \(y = \frac{x^2}{9} + \frac{z^2}{9} - 1\) with \(z \leq 0\) oriented so that the normal vector S has a positive y component. Use Stokes’ theorem to compute integral \[\iint_S curl \, F \cdot dS.\]

Use Stokes’ theorem to evaluate \[ \oint F \cdot dS,\] where \(F(x,y,z) = yi + zj + xk\) and C is a triangle with vertices (0, 0, 0), (2, 0, 0) and \(0,-2,2)\) oriented counterclockwise when viewed from above.

[Hide Solution]

\[ \oint F \cdot dS = -4\]

 

Use the surface integral in Stokes’ theorem to calculate the circulation of field F, \(F(x,y,z) = x^2y^3 i + j + zk\) around C, which is the intersection of cylinder \(x^2 + y^2 = 4\) and hemisphere \(x^2 + y^2 + z^2 = 16, \, z \geq 0\), oriented counterclockwise when viewed from above.

A diagram in three dimensions of a vector field and the intersection of a sylinder and hemisphere. The arrows are horizontal and have negative x components for negative y components and have positive x components for positive y components. The curve of intersection between the hemisphere and cylinder is drawn in blue.

Use Stokes’ theorem to compute \[\iint_S curl \, F \cdot dS.\] where \(F(x,y,z) = i + xy^2j + xy^2 k\) and S is a part of plane \(y + z = 2\)  inside cylinder \(x^2 + y^2 = 1\) and oriented counterclockwise.

A diagram of a vector field in three dimensional space showing the intersection of a plane and a cylinder. The curve where the plane and cylinder intersect is drawn in blue.

[Hide Solution]

\[\iint_S curl \, F \cdot dS = 0\]

 

Use Stokes’ theorem to evaluate  \[\iint_S curl \, F \cdot dS,\] where \(F(x,y,z) = -y^2 i + xj + z^2k\) and S is the part of plane \(x + y + z = 1\) in the positive octant and oriented counterclockwise \(x \geq 0, \, y \geq 0, \, z \geq 0\).

Let \(F(x,y,z) = xyi + 2zj - 2yk\) and let C be the intersection of plane \(x + z = 5\) and cylinder \(x^2 + y^2 = 9\), which is oriented counterclockwise when viewed from the top. Compute the line integral of F over C using Stokes’ theorem.

[Hide Solution]

\[\iint_S curl \, F \cdot dS = -36 \pi\]

[T] Use a CAS and let \(F(x,y,z) = xy^2i + (yz - x)j + e^{yxz}k\). Use Stokes’ theorem to compute the surface integral of curl F over surface S with inward orientation consisting of cube \([0,1] \times [0,1] \times [0,1]\) with the right side missing.

 

Let S be ellipsoid \(\frac{x^2}{4} + \frac{y^2}{9} + z^2 = 1\) oriented counterclockwise and let F be a vector field with component functions that have continuous partial derivatives.

[Hide Solution]

\[\iint_S curl \, F \cdot N = 0\]

 

Let S be the part of paraboloid \(z = 9 - x^2 - y^2\) with \(z \geq 0\). Verify Stokes’ theorem for vector field \(F(x,y,z) = 3zi + 4xj + 2yk\).

[T] Use a CAS and Stokes’ theorem to evaluate \[\oint F \cdot dS,\] if \(F(x,y,z) = (3z - \sin x) i + (x^2 + e^y) j + (y^3 - \cos z) k\), where C is the curve given by \(x = \cos t, \, y = \sin t, \, z = 1; \, 0 \leq t \leq 2\pi\).

[Hide Solution]

\[\oint_C F \cdot dr = 0\]

 

[T] Use a CAS and Stokes’ theorem to evaluate \(F(x,y,z) = 2yi + e^zj - arctan \, xk\) with S as a portion of paraboloid \(z = 4 - x^2 - y^2\) cut off by the xy-plane oriented counterclockwise.

[T] Use a CAS to evaluate \[\iint_S curl (F) \cdot dS,\] where \(F(x,y,z) = 2zi + 3xj + 5yk\) and S is the surface parametrically by \(r(r,\theta) = r \, \cos \theta i + r \, \sin \theta j + (4 - r^2) k \, (0 \leq \theta \leq 2\pi, \, 0 \leq r \leq 3)\).

[Hide Solution]

\[\iint_S curl (F) \cdot dS = 84.8230\]

 

Let S be paraboloid \(z = a (1 - x^2 - y^2)\), for \(z \geq 0\), where \(a > 0\) is a real number. Let \(F(x,y,z) = \langle x - y, \, y + z, \, z - x \rangle\). For what value(s) of a (if any) does \[\iint_S (\nabla \times F) \cdot n \, dS\] have its maximum value?

 

For the following application exercises, the goal is to evaluate \[A = \iint_S (\nabla \times F) \cdot n \, dS,\] where \(F = \langle xz, \, -xz, \, xy \rangle\) and S is the upper half of ellipsoid \(x^2 + y^2 + 8z^2 = 1\), where \(z \geq 0\).

Evaluate a surface integral over a more convenient surface to find the value of A.

[Hide Solution]

\[A = \iint_S (\nabla \times F) \cdot n \, dS = 0\]

Evaluate A using a line integral.

Take paraboloid \(z = x^2 + y^2\), for \(0 \leq z \leq 4\), and slice it with plane \(y = 0\). Let S be the surface that remains for \(y \geq 0\), including the planar surface in the xz-plane. Let C be the semicircle and line segment that bounded the cap of S in plane \(z = 4\) with counterclockwise orientation. Let \(F = \langle 2z + y, \, 2x + z, \, 2y + x \rangle\). Evaluate \[\iint_S (\nabla \times F) \cdot n \, dS.\]

A diagram of a vector field in three dimensional space where a paraboloid with vertex at the origin, plane at y=0, and plane at z=4 intersect. The remaining surface is the half of a paraboloid under z=4 and above y=0.

[Hide Solution]

\[\iint_S (\nabla \times F) \cdot n \, dS = 2\pi\]

 

For the following exercises, let S be the disk enclosed by curve \(C \, : \, r(t) = \langle \cos \varphi \, \cos t, \, \sin t, \, \sin \varphi \, \cos t \rangle\), for \(0 \leq t \leq 2\pi\), where \(0 \leq \varphi \leq \frac{\pi}{2}\) is a fixed angle.

What is the length of C in terms of \(\varphi\)?

What is the circulation of C of vector field \(F = \langle -y, \, -z, \, x \rangle\) as a function of \(\varphi\)?

[Hide Solution]

 

\(C = \pi (\cos \varphi - \sin \varphi)\)

For what value of \(\varphi\) is the circulation a maximum?

 

Circle C in plane \(x + y + z = 8\) has radius 4 and center (2, 3, 3). Evaluate \[\oint_C F \cdot dr\] for \(F = \langle 0, \, -z, \, 2y \rangle\), where C has a counterclockwise orientation when viewed from above.

[Hide Solution]

\[\oint_C F \cdot dr = 48 \pi\]

Velocity field \(v = \langle 0, \, 1 -x^2, \, 0 \rangle \), for \(|x| \leq 1\) and \(|z| \leq 1\), represents a horizontal flow in the y-direction. Compute the curl of v in a clockwise rotation.

Evaluate integral \[ \iint_S (\nabla \times F) \cdot n \, dS,\] where \(F = - xzi + yzj + xye^z k\) and S is the cap of paraboloid \(z = 5 - x^2 - y^2\) above plane \(z = 3\), and n points in the positive z-direction on S.

[Hide Solution]

\[ \iint_S (\nabla \times F) \cdot n = 0\]

For the following exercises, use Stokes’ theorem to find the circulation of the following vector fields around any smooth, simple closed curve C.

\(F = \nabla (x \, \sin ye^z)\)

\(F = \langle y^2z^3, \, z2xyz^3, 3xy^2z^2 \rangle \)

[Hide Solution]

0