Lesson 34: Stokes Theorem - I (17.7)

Accessible transcription generated on 4/13/2026

Original Notes

Page 1

Lesson 34: Stokes Theorem - I (17.7)

Review example: Evaluate \(\iint_S \vec{F} \cdot d\vec{S}\), where \(\vec{F} = 4\vec{k} = \langle 0, 0, 4 \rangle\) and \(S\) is the rectangle in the \(xy\)-plane. \[ \iint_S \vec{F} \cdot d\vec{S} = \iint_S \vec{F} \cdot \vec{n} \, dS \]
Convention: The unit normal vector \(\vec{n}\) should point up or outward.
A 3D coordinate system with x, y, and z axes. A green shaded rectangular surface labeled S lies in the xy-plane. The rectangle's boundaries are defined by its vertices, including (3, 0, 0) on the x-axis and (0, 2, 0) on the y-axis. Two blue arrows, representing the unit normal vector n, point vertically upwards from the surface S in the positive z-direction.
Visual Description: A 3D coordinate system with x, y, and z axes. A green shaded rectangular surface labeled S lies in the xy-plane. The rectangle's boundaries are defined by its vertices, including (3, 0, 0) on the x-axis and (0, 2, 0) on the y-axis. Two blue arrows, representing the unit normal vector n, point vertically upwards from the surface S in the positive z-direction.
In the \(xy\)-plane \(\Rightarrow z = 0\). The unit normal vector is \(\vec{n} = \vec{k} = \langle 0, 0, 1 \rangle\). \[ \iint_S \vec{F} \cdot d\vec{S} = \iint_S \vec{F} \cdot \vec{n} \, dS = \iint_S 4 \, dS \] \[ = 4 \iint_S 1 \, dS \] \[ = 4 \times \text{Area}(S) \] \[ = 4 \times 6 = 24 \]

Suppose \(\vec{G} = 7\vec{j} = \langle 0, 7, 0 \rangle\), then:

\[ \iint_S \vec{G} \cdot d\vec{S} = \iint_S \vec{G} \cdot \vec{n} \, dS = 0 \] (Since \(\vec{G} \cdot \vec{n} = \langle 0, 7, 0 \rangle \cdot \langle 0, 0, 1 \rangle = 0\))

Page 2

Surface Integrals

A diagram illustrating the parameterization of a surface. On the left, a 2D coordinate system with axes labeled u and v contains a blue-outlined region R. A red arrow labeled r(u, v) points from region R to a green-shaded surface S in a 3D coordinate system (represented by a vertical z-axis) on the right. A purple normal vector n is shown pointing outward from the surface S. Next to surface S, the formula for the unit normal vector is given as \vec{n} = \pm \frac{(r_u \times r_v)}{|r_u \times r_v|}.
Visual Description: A diagram illustrating the parameterization of a surface. On the left, a 2D coordinate system with axes labeled u and v contains a blue-outlined region R. A red arrow labeled r(u, v) points from region R to a green-shaded surface S in a 3D coordinate system (represented by a vertical z-axis) on the right. A purple normal vector n is shown pointing outward from the surface S. Next to surface S, the formula for the unit normal vector is given as \vec{n} = \pm \frac{(r_u \times r_v)}{|r_u \times r_v|}.
The surface integral of a scalar function \( f \) over a surface \( S \) is defined by mapping it back to the region \( R \) in the \( uv \)-plane: \[ \iint_S f \, dS = \iint_R f(r(u,v)) |r_u \times r_v| \, dA \] The surface integral of a vector field \( \vec{F} \) over an oriented surface \( S \), also known as the flux, is defined as: \[ \iint_S \vec{F} \cdot d\vec{S} = \iint_S \vec{F} \cdot \vec{n} \, dS = \iint_R \vec{F}(r(u,v)) \cdot (\pm r_u \times r_v) \, dA \]
Regarding the \(\pm\) sign in the vector surface integral formula: Choose \(\pm\) based on what's given or convention.

Page 3

Stokes Theorem \(\rightarrow\) extension of Green's to 3D

Green's Theorem: \(C\) is a 2D closed curve.

\[ \vec{F} = \langle f(x,y), g(x,y) \rangle \]
A 3D coordinate system diagram with labeled x, y, and z axes. The y-axis is vertical (pointing up), the x-axis is horizontal (pointing right), and the z-axis is drawn diagonally (pointing down and to the left). In the xy-plane, there is a shaded region labeled 'S', which is enclosed by a green boundary curve labeled 'C'. The region 'S' is filled with purple diagonal hatching lines. This visualizes the planar region and its boundary in a 3D context.
Visual Description: A 3D coordinate system diagram with labeled x, y, and z axes. The y-axis is vertical (pointing up), the x-axis is horizontal (pointing right), and the z-axis is drawn diagonally (pointing down and to the left). In the xy-plane, there is a shaded region labeled 'S', which is enclosed by a green boundary curve labeled 'C'. The region 'S' is filled with purple diagonal hatching lines. This visualizes the planar region and its boundary in a 3D context.
\[ \oint_C \vec{F} \cdot d\vec{r} = \iint_S (g_x - f_y) dA \]

In this expression, the term \( (g_x - f_y) \) is the 2D curl.

Extend to 3D:

To extend the theorem to three dimensions, we define the vector field \(\vec{F}\) as:

\[ \vec{F} = \langle f(x,y), g(x,y), 0 \rangle \]

Consider \(S\) as a region in the \(xy\) plane, with boundary curve \(C\) and a normal vector \(\vec{n} = \langle 0, 0, 1 \rangle\).

The curl of the vector field is:

\[ \text{curl } \vec{F} = \nabla \times \vec{F} = \langle 0, 0, g_x - f_y \rangle \]

Using these definitions, the relationship can be expressed as:

\[ \oint_C \vec{F} \cdot d\vec{r} = \iint_S \text{curl } \vec{F} \cdot \vec{n} \, dA \]

Page 4

Stokes' Theorem and Orientation

A technical diagram illustrating an oriented surface S and its boundary curve C. The surface S is shown as a 3D dome-like shape with purple diagonal hatching. A blue arrow points vertically upwards from the surface, representing the positive unit normal vector field. The boundary of this surface is a closed green curve C at the base. A green arrow on curve C indicates a counter-clockwise orientation when viewed from above, which is consistent with the right-hand rule relative to the upward-pointing surface normal.
Visual Description: A technical diagram illustrating an oriented surface S and its boundary curve C. The surface S is shown as a 3D dome-like shape with purple diagonal hatching. A blue arrow points vertically upwards from the surface, representing the positive unit normal vector field. The boundary of this surface is a closed green curve C at the base. A green arrow on curve C indicates a counter-clockwise orientation when viewed from above, which is consistent with the right-hand rule relative to the upward-pointing surface normal.

If \( S \) is an oriented surface with boundary an oriented curve \( C \), then:

\[ \oint_{C} \vec{F} \cdot d\vec{r} = \iint_{S} \text{curl } \vec{F} \cdot d\vec{S} \]

Orientation of \( S \) is determined by its normal vector.

For curve \( C \):

Two simplified diagrams of closed loops showing possible directions for a curve C. The first loop on the left has a green arrow indicating a clockwise orientation. The word 'or' is written in green between the two loops. The second loop on the right has a green arrow indicating a counter-clockwise orientation.
Visual Description: Two simplified diagrams of closed loops showing possible directions for a curve C. The first loop on the left has a green arrow indicating a clockwise orientation. The word 'or' is written in green between the two loops. The second loop on the right has a green arrow indicating a counter-clockwise orientation.

Choose direction/orientation such that when you roll your fingers in the direction of curve \( C \), your thumb points in the normal direction of \( S \).

Page 5

Example

\( \vec{F} = \langle y, -x, 0 \rangle \)
\( S \) is part of sphere of Radius 2, \( z \geq 1 \). Normal pointing outward.

Normal pointing outward \( \Downarrow \)
z coordinate is \( \geq 0 \)

Find \( \iint_S \text{curl} \vec{F} \cdot d\vec{S} \)

A 3D coordinate diagram showing a spherical cap S situated above the xy-plane. The cap is the part of a sphere of radius 2 where z is greater than or equal to 1. Blue arrows represent outward-pointing normal vectors on the surface S. The boundary of the surface is a circular curve C in a horizontal plane. A green arrow on curve C indicates a counter-clockwise orientation when viewed from above. The x, y, and z axes are shown originating from (0,0,0) below the cap.
Visual Description: A 3D coordinate diagram showing a spherical cap S situated above the xy-plane. The cap is the part of a sphere of radius 2 where z is greater than or equal to 1. Blue arrows represent outward-pointing normal vectors on the surface S. The boundary of the surface is a circular curve C in a horizontal plane. A green arrow on curve C indicates a counter-clockwise orientation when viewed from above. The x, y, and z axes are shown originating from (0,0,0) below the cap.

Two ways to compute:

  1. Direct:
    • Find \( \text{curl} \vec{F} \)
    • Parametrize \( S: \vec{r}(u,v) \) on \( R \)
    • Find \( \vec{r}_u \times \vec{r}_v \)
    • Evaluate: \[ \iint_S \text{curl} \vec{F} \cdot d\vec{S} = \iint_R \text{curl} \vec{F} \cdot (\pm \vec{r}_u \times \vec{r}_v) \, dA \]
  2. Stokes:
    • Parametrize \( C: \vec{r}(t), a \leq t \leq b \)
    • Evaluate: \[ \iint_S \text{curl} \vec{F} \cdot d\vec{S} = \oint_C \vec{F} \cdot d\vec{r} = \int_a^b \vec{F}(\vec{r}(t)) \cdot \vec{r}'(t) \, dt \]

Page 6

Vector Field Calculation and Surface Parametrization

A 3D Cartesian coordinate system featuring a surface S and its boundary curve C. The surface S is a hemispherical dome portion, shaded with purple hatch lines, situated above the xy-plane. A blue arrow points away from the surface, indicating its orientation. The boundary of the dome is a green circular curve C, which includes an arrow indicating a counter-clockwise orientation when viewed from above. Labels 'S' and 'C' are placed near their respective components.
Visual Description: A 3D Cartesian coordinate system featuring a surface S and its boundary curve C. The surface S is a hemispherical dome portion, shaded with purple hatch lines, situated above the xy-plane. A blue arrow points away from the surface, indicating its orientation. The boundary of the dome is a green circular curve C, which includes an arrow indicating a counter-clockwise orientation when viewed from above. Labels 'S' and 'C' are placed near their respective components.

Direct:

\[ \text{curl } \vec{F} = \vec{\nabla} \times \vec{F} = \begin{vmatrix} \bar{i} & \bar{j} & \bar{k} \\ \partial/\partial x & \partial/\partial y & \partial/\partial z \\ y & -x & 0 \end{vmatrix} \] \[ = \langle 0, 0, -2 \rangle \]

Parametrize \( S \): Use spherical coordinates

\[ \begin{aligned} x &= \rho \sin \phi \cos \theta \\ y &= \rho \sin \phi \sin \theta \\ z &= \rho \cos \phi \end{aligned} \]

\( \rho = 2 \leadsto \) sphere of radius 2
\( \phi \) is 0 to when you hit \( z = 1 \leadsto 1 = 2 \cos \phi \leadsto \cos \phi = 1/2 \Rightarrow \phi = \pi/3 \)
\( 0 \le \theta \le 2\pi \)

Page 7

Surface Parametrization and Vector Normal Calculation

\[ r(\phi, \theta) = \langle 2 \sin\phi \cos\theta, 2 \sin\phi \sin\theta, 2 \cos\phi \rangle \] \[ 0 \leq \phi \leq \pi/3 \] \[ 0 \leq \theta \leq 2\pi \]

Partial derivatives with respect to \(\phi\) and \(\theta\): \[ r_{\phi} = \langle 2 \cos\phi \cos\theta, 2 \cos\phi \sin\theta, -2 \sin\phi \rangle \] \[ r_{\theta} = \langle -2 \sin\phi \sin\theta, 2 \sin\phi \cos\theta, 0 \rangle \]

Calculating the cross product to find the normal vector: \[ r_{\phi} \times r_{\theta} = \langle 4 \sin^{2}\phi \cos\theta, 4 \sin^{2}\phi \sin\theta, 4 \cos\phi \sin\phi (\underbrace{\cos^{2}\theta + \sin^{2}\theta}_{1}) \rangle \]

Evaluating the orientation based on the \(z\)-coordinate: \[ z \text{-coordinate} = 4 \cos\phi \sin\phi > 0 \] for \( 0 < \phi < \pi/3 \)

So we choose \( r_{\phi} \times r_{\theta} \) instead of \( -r_{\phi} \times r_{\theta} \).

Page 8

Evaluation of the Surface Integral

\[ \iint_{S} \text{curl } \vec{F} \cdot d\vec{S} = \int_{0}^{2\pi} \int_{0}^{\pi/3} \text{curl } \vec{F} \cdot (\vec{r}_{\phi} \times \vec{r}_{\theta}) \, d\phi \, d\theta \] \[ = \int_{0}^{2\pi} \int_{0}^{\pi/3} \langle 0, 0, -2 \rangle \cdot \langle \, , \, , 4 \cos \phi \sin \phi \rangle \, d\phi \, d\theta \] \[ = -8 \int_{0}^{2\pi} \int_{0}^{\pi/3} \cos \phi \sin \phi \, d\phi \, d\theta \] \[ = -8 * 2\pi * \left. \frac{\sin^2 \phi}{2} \right|_{0}^{\pi/3} = -8 \times \pi \times \left[ \frac{\sqrt{3}}{2} \right]^2 \] \[ = -6\pi \]

Page 9

Using Stokes:

A 3D coordinate system diagram illustrating the setup for Stokes' Theorem. It shows a three-dimensional hemispherical surface labeled \( S \), shaded with pink diagonal lines, and several blue arrows representing normal vectors pointing outwards. The base of the dome is a horizontal boundary curve labeled \( C \), shown in green. The coordinate system includes a vertical z-axis and two horizontal axes (x and y). The curve \( C \) is situated above the xy-plane, representing the intersection of the sphere and a horizontal plane.
Visual Description: A 3D coordinate system diagram illustrating the setup for Stokes' Theorem. It shows a three-dimensional hemispherical surface labeled \( S \), shaded with pink diagonal lines, and several blue arrows representing normal vectors pointing outwards. The base of the dome is a horizontal boundary curve labeled \( C \), shown in green. The coordinate system includes a vertical z-axis and two horizontal axes (x and y). The curve \( C \) is situated above the xy-plane, representing the intersection of the sphere and a horizontal plane.
\( C \): intersection of \( z = 1 \) & \( x^2 + y^2 + z^2 = 4 \)
\( \Rightarrow x^2 + y^2 = 3, \quad z = 1 \)
\( C \): circle of radius \( \sqrt{3} \) at \( z = 1 \)

\( r(t) = \langle \sqrt{3} \cos t, \sqrt{3} \sin t, 1 \rangle \)
\( 0 \le t \le 2\pi \)

\( r'(t) = \langle -\sqrt{3} \sin t, \sqrt{3} \cos t, 0 \rangle \)
\( F(r(t)) = \langle \sqrt{3} \sin t, -\sqrt{3} \cos t, 0 \rangle \)

\[ \iint_{S} \text{curl } \vec{F} \cdot d\vec{S} = \oint_{C} \vec{F} \cdot d\vec{r} = \int_{0}^{2\pi} F(r(t)) \cdot r'(t) \, dt = \int_{0}^{2\pi} (-3) \, dt = -6\pi \]

Page 10

Stokes' Theorem Example

Example: Given the vector field \(\vec{F} = \langle xz - y, x - yz, x^2 \rangle\) and a surface \(S\) that is the part of the paraboloid \(z = 1 - x^2 - y^2\) above the \(xy\)-plane, with the normal pointing upward, find: \[ \iint_{S} \text{curl } \vec{F} \cdot d\vec{S} \]
A 3D coordinate system diagram showing the surface S. The surface is a downward-opening paraboloid defined by \(z = 1 - x^2 - y^2\), located above the xy-plane. The peak of the paraboloid is at (0, 0, 1) on the vertical z-axis, and its base is a circle of radius 1 centered at the origin in the xy-plane. Blue arrows on the surface point upward and outward, representing the positive orientation of the normal vectors. The x, y, and z axes are drawn to show the spatial orientation.
Visual Description: A 3D coordinate system diagram showing the surface S. The surface is a downward-opening paraboloid defined by \(z = 1 - x^2 - y^2\), located above the xy-plane. The peak of the paraboloid is at (0, 0, 1) on the vertical z-axis, and its base is a circle of radius 1 centered at the origin in the xy-plane. Blue arrows on the surface point upward and outward, representing the positive orientation of the normal vectors. The x, y, and z axes are drawn to show the spatial orientation.

By Stokes' Theorem, the surface integral of the curl is equal to the line integral around the boundary curve \(C\):

\[ \iint_{S} \text{curl } \vec{F} \cdot d\vec{S} = \oint_{C} \vec{F} \cdot d\vec{r} \]

First, we identify the boundary curve \(C\), which is the intersection of the paraboloid and the \(xy\)-plane:

  • Surface equation: \(z = 1 - x^2 - y^2\)
  • \(xy\)-plane equation: \(z = 0\)

Setting \(z = 0\), we get \(0 = 1 - x^2 - y^2\), which is a circle of radius 1 at \(z = 0\).

Next, we parameterize the curve \(C\):

\[ \vec{r}(t) = \langle \cos t, \sin t, 0 \rangle, \quad 0 \le t \le 2\pi \]

The derivative is:

\[ \vec{r}'(t) = \langle -\sin t, \cos t, 0 \rangle \]

Evaluating the vector field \(\vec{F}\) on the curve \(\vec{r}(t)\):

\[ \vec{F}(\vec{r}(t)) = \langle xz - y, x - yz, x^2 \rangle = \langle (\cos t)(0) - \sin t, \cos t - (\sin t)(0), \cos^2 t \rangle \] \[ \vec{F}(\vec{r}(t)) = \langle -\sin t, \cos t, \cos^2 t \rangle \]

Finally, we calculate the line integral:

\[ \iint_{S} \text{curl } \vec{F} \cdot d\vec{S} = \oint_{C} \vec{F} \cdot d\vec{r} = \int_{0}^{2\pi} \vec{F}(\vec{r}(t)) \cdot \vec{r}'(t) \, dt \] \[ = \int_{0}^{2\pi} \langle -\sin t, \cos t, \cos^2 t \rangle \cdot \langle -\sin t, \cos t, 0 \rangle \, dt \] \[ = \int_{0}^{2\pi} (\sin^2 t + \cos^2 t + 0) \, dt \] \[ = \int_{0}^{2\pi} 1 \, dt = [t]_{0}^{2\pi} = \mathbf{2\pi} \]

Page 11

Application of Stokes' Theorem

A 3D coordinate diagram illustrating the relationship between two surfaces sharing the same boundary curve for the application of Stokes' Theorem. A purple dome-shaped surface, labeled S, rises from the xy-plane into the positive z-direction. Its base is a green closed curve, labeled C, which lies entirely in the xy-plane. Several blue arrows represent the outward-pointing normal vectors of surface S. Within the same green boundary curve C, a second surface labeled S2 is shown as a flat, red-hashed disk lying in the xy-plane. The z-axis is indicated by a vertical blue arrow, and the x and y axes are represented by black lines.
Visual Description: A 3D coordinate diagram illustrating the relationship between two surfaces sharing the same boundary curve for the application of Stokes' Theorem. A purple dome-shaped surface, labeled S, rises from the xy-plane into the positive z-direction. Its base is a green closed curve, labeled C, which lies entirely in the xy-plane. Several blue arrows represent the outward-pointing normal vectors of surface S. Within the same green boundary curve C, a second surface labeled S2 is shown as a flat, red-hashed disk lying in the xy-plane. The z-axis is indicated by a vertical blue arrow, and the x and y axes are represented by black lines.

\( C \) is also the boundary for \( S_2 \).

\[ \iint_S \text{curl } \vec{F} \cdot d\vec{S} = \oint_C \vec{F} \cdot d\vec{r} \] \[ = \iint_{S_2} \text{curl } \vec{F} \cdot d\vec{S} \]
Another way to apply Stokes': find an easy surface with the same boundary.