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17calculus > vector fields > surface integrals |
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Surface Orientation |
Surface Integrals of Vector Fields |
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There are several equations for surface integrals and which one you use depends on what form your equations are in. This is a similar situation that you encountered with line integrals.
There are two main groups of equations, one for surface integrals of scalar-valued functions and a second group for surface integrals of vector fields (often called flux integrals). The following table places them side-by-side so that you can easily see the difference.
scalar-valued function |
\(f(x,y,z)\) | |
vector field |
\(\vec{F}(x,y,z)=M(x,y,z)\hat{i} + N(x,y,z)\hat{j} + P(x,y,z)\hat{k}\) |
What Are Surface Integrals? |
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Before we get started with the details of surface integrals and how to evaluate them, let's watch a couple of great videos that will gently introduce you to surface integrals of scalar-valued functions and give you some examples. This is one of our favorite instructors and we think these videos are worth taking the time to watch.
video by Dr Chris Tisdell
In this second video from about the 15min-20sec mark to the end of the video, he discusses a way to simplify the integration in a special case. We suggest that you do not watch that part of the video until you have worked some practice problems and are comfortable with the basics of setting up the integrals. However, check with your instructor to see if they will allow you to use this technique.
video by Dr Chris Tisdell
Surface Integrals of Scalar-Valued Functions |
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The following equations are used when you are given a scalar-valued function over which you need to evaluate a surface integral. The form of the function is \(f(x,y,z)\). In general, your surface is parameterized as \(\vec{r}(u,v)=x(u,v)\hat{i} + y(u,v)\hat{j} + z(u,v)\hat{k}\). So to evaluate the integral of \(f(x,y,z)\) over the surface \(\vec{r}\), we use the equation
\( \iint\limits_S {f(x,y) ~ dS} = \iint\limits_R {f(x(u,v),y(u,v),z(u,v)) ~ \| \vec{r}_u \times \vec{r}_v \| ~ dA} \) |
\(\vec{r}(u,v)\) is the parametric surface |
R is the region in the uv-plane |
\(\vec{r}_u\) and \(\vec{r}_v\) are the partial derivatives of \(\vec{r}\) |
In the special case where we have the surface described as \(z=g(x,y)\), we can parameterize the surface as \(\vec{r}=x\hat{i}+y\hat{j}+g(x,y)\hat{k}\). This gives \( \| \vec{r}_x \times \vec{r}_y \| = \sqrt{1+[g_x]^2+[g_y]^2} \) and the surface integral can then be written as \( \iint\limits_S {f(x,y) ~ dS} = \iint\limits_R {f(x,y,z) ~ \sqrt{1+[g_x]^2+[g_y]^2} ~ dA} \) and R is the region in the xy-plane.
Okay, let's try some practice problems evaluating surface integrals of scalar functions.
Basic Problems |
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Evaluate \( \iint_S { x^2 y z ~ dS } \) where S is the part of the plane \( z = 1 + 2x + 3y \) that lies above the rectangle \( 0 \leq x \leq 3, 0 \leq y \leq 2 \).
Problem Statement |
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Evaluate \( \iint_S { x^2 y z ~ dS } \) where S is the part of the plane \( z = 1 + 2x + 3y \) that lies above the rectangle \( 0 \leq x \leq 3, 0 \leq y \leq 2 \).
Final Answer |
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\( 171 \sqrt{14} \) |
Problem Statement |
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Evaluate \( \iint_S { x^2 y z ~ dS } \) where S is the part of the plane \( z = 1 + 2x + 3y \) that lies above the rectangle \( 0 \leq x \leq 3, 0 \leq y \leq 2 \).
Solution |
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video by PatrickJMT
Final Answer |
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\( 171 \sqrt{14} \) |
close solution |
Evaluate \( \iint_S{ xy ~ dS } \) using a parametric surface where S is \( x^2 + y^2 = 4, 0 \leq z \leq 8 \) in the first octant.
Problem Statement |
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Evaluate \( \iint_S{ xy ~ dS } \) using a parametric surface where S is \( x^2 + y^2 = 4, 0 \leq z \leq 8 \) in the first octant.
Final Answer |
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32 |
Problem Statement |
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Evaluate \( \iint_S{ xy ~ dS } \) using a parametric surface where S is \( x^2 + y^2 = 4, 0 \leq z \leq 8 \) in the first octant.
Solution |
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video by MIP4U
Final Answer |
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32 |
close solution |
Evaluate \( \iint_S { x^2 + y^2 } \) using a parametric surface where S is the hemisphere \( x^2 + y^2 + z^2 = 1 \) above the xy-plane.
Problem Statement |
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Evaluate \( \iint_S { x^2 + y^2 } \) using a parametric surface where S is the hemisphere \( x^2 + y^2 + z^2 = 1 \) above the xy-plane.
Final Answer |
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\(4\pi/3\) |
Problem Statement |
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Evaluate \( \iint_S { x^2 + y^2 } \) using a parametric surface where S is the hemisphere \( x^2 + y^2 + z^2 = 1 \) above the xy-plane.
Solution |
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video by MIP4U
Final Answer |
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\(4\pi/3\) |
close solution |
Integrate \( f(x,y,z) = xy \) over the surface \( z = 4 - 2x - 2y \) in the first octant.
Problem Statement |
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Integrate \( f(x,y,z) = xy \) over the surface \( z = 4 - 2x - 2y \) in the first octant.
Final Answer |
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2 |
Problem Statement |
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Integrate \( f(x,y,z) = xy \) over the surface \( z = 4 - 2x - 2y \) in the first octant.
Solution |
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video by MIP4U
Final Answer |
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2 |
close solution |
Intermediate Problems |
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Compute the surface integral of \(\displaystyle{ f(x,y,z) = \frac{2z^2}{x^2+y^2+z^2} }\) over the cap of the sphere \( x^2 + y^2 + z^2 = 9 \), \(z \geq 2\).
Problem Statement |
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Compute the surface integral of \(\displaystyle{ f(x,y,z) = \frac{2z^2}{x^2+y^2+z^2} }\) over the cap of the sphere \( x^2 + y^2 + z^2 = 9 \), \(z \geq 2\).
Final Answer |
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\( 76 \pi/9 \) |
Problem Statement |
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Compute the surface integral of \(\displaystyle{ f(x,y,z) = \frac{2z^2}{x^2+y^2+z^2} }\) over the cap of the sphere \( x^2 + y^2 + z^2 = 9 \), \(z \geq 2\).
Solution |
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Note - At about the 13min-50sec mark, he writes \(76\pi/3\) as the answer. At the end of the video, he corrects his answer to \(76\pi/9\).
video by Dr Chris Tisdell
Final Answer |
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\( 76 \pi/9 \) |
close solution |
A roof is given by the graph of \( g(x,y) = 25 + 0.5x + 0.5y \) over \( 0 \leq x \leq 40 \), \( 0 \leq y \leq 20 \). If the density of the roof is given by \( f(x,y,z) = 150 - 2z \), determine the mass of the roof.
Problem Statement |
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A roof is given by the graph of \( g(x,y) = 25 + 0.5x + 0.5y \) over \( 0 \leq x \leq 40 \), \( 0 \leq y \leq 20 \). If the density of the roof is given by \( f(x,y,z) = 150 - 2z \), determine the mass of the roof.
Final Answer |
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\( 28000\sqrt{6} \) |
Problem Statement |
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A roof is given by the graph of \( g(x,y) = 25 + 0.5x + 0.5y \) over \( 0 \leq x \leq 40 \), \( 0 \leq y \leq 20 \). If the density of the roof is given by \( f(x,y,z) = 150 - 2z \), determine the mass of the roof.
Solution |
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video by MIP4U
Final Answer |
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\( 28000\sqrt{6} \) |
close solution |
Surface Orientation |
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Before we discuss surface integrals over vector fields, we need to discuss surface orientation. Surface orientation is important because we need to know which direction the vector field is pointing, outside or inside, in order to determine the flux through the surface.
The two vectors that calculated above, \(\vec{r}_u\) and \(\vec{r}_v\) and tangent vectors to the surface. Using the cross product, we can calculate two possible normal vectors, \(\vec{N}_1 = \vec{r}_u \times \vec{r}_v\) and \(\vec{N}_2 = \vec{r}_v \times \vec{r}_u\). One vector points inward, the other points outward. We will use the first one, i.e. \(\vec{N}_1 = \vec{r}_u \times \vec{r}_v\) and divide by the length to get the unit vector \(\displaystyle{ \vec{N} = \frac{\vec{r}_u \times \vec{r}_v}{ \| \vec{r}_u \times \vec{r}_v \|} }\) which is called the upward unit normal.
[Important Note: This may not always be the outward pointing normal but for our discussions we will work with surfaces where the outward pointing normal is this upward pointing normal. For your application, you will need to double check that you have the outward pointing normal.]
Surface Integrals of Vector Fields |
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In this video, Dr Chris Tisdell continues his discussion of surface integrals and talks about vector fields. Again, this is a great video to watch.
video by Dr Chris Tisdell
Surface integrals over vector fields are often called flux integrals since we will often be calculating the flux through a closed surface. The flux of a vector field \(\vec{F}(x,y,z)=M(x,y,z)\hat{i}+N(x,y,z)\hat{j}+P(x,y,z)\hat{k}\) through a surface S with a unit normal vector \(\vec{N}\) is \( \iint\limits_S { \vec{F} \cdot \vec{N} ~ dS} \).
[ Note: In the above description, there are two N's. One is a function \(N(x,y,z)\) which is the j-component of the vector function. The other is \(\vec{N}\), a unit normal vector. They are distinct and unrelated and should not be confused. ]
When the surface is given in terms of \(z=g(x,y)\), we can calculate the unit normal vector as \(\displaystyle{ \frac{\nabla G}{\| \nabla G \|} }\). Since \(dS = \| \nabla G \| ~ dA \) where \(G(x,y,z)=z-g(x,y)\), the surface integral becomes \( \iint\limits_R { \vec{F} \cdot \nabla G ~ dA } \) where R is the projection of S in the xy-plane.
Surface Integrals - Meaning and Applications |
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The meaning of the surface integral depends on what the function \(f(x,y,z)\) or \(\vec{F}(x,y,z)\) represents. Here is a video clip giving some applications.
video by Evans Lawrence
In this final video, he gives more explanation of surface integrals and a couple of examples. He has a unique way of thinking about surface integrals and, as he says at the first of this video, surface integration is not easy. So it will help you to watch this video clip as well before going on to trying some on your own.
video by Evans Lawrence
Okay, you are now ready for some practice problems calculating surface integrals using vector functions.
Then you will be ready for the three dimensional versions of Green's Theorem, Stokes' Theorem and the Divergence Theorem.
Basic Problems |
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Compute the flux of \( \vec{F} = \langle x,y,z \rangle \) across the surface \( z = 4 - x^2 - y^2 \), \( z \geq 0 \) oriented up.
Problem Statement |
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Compute the flux of \( \vec{F} = \langle x,y,z \rangle \) across the surface \( z = 4 - x^2 - y^2 \), \( z \geq 0 \) oriented up.
Final Answer |
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\( 24\pi \) |
Problem Statement |
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Compute the flux of \( \vec{F} = \langle x,y,z \rangle \) across the surface \( z = 4 - x^2 - y^2 \), \( z \geq 0 \) oriented up.
Solution |
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video by Evans Lawrence
Final Answer |
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\( 24\pi \) |
close solution |
Determine the flux of \( \vec{F} = \langle 0,-1,-2 \rangle \) across the surface \( z = 6 - x - y \) in the first octant. Use a downward orientation.
Problem Statement |
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Determine the flux of \( \vec{F} = \langle 0,-1,-2 \rangle \) across the surface \( z = 6 - x - y \) in the first octant. Use a downward orientation.
Final Answer |
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54 |
Problem Statement |
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Determine the flux of \( \vec{F} = \langle 0,-1,-2 \rangle \) across the surface \( z = 6 - x - y \) in the first octant. Use a downward orientation.
Solution |
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video by MIP4U
Final Answer |
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54 |
close solution |
Determine the surface area of the cylinder given by \( \vec{r} = \langle 3\cos(u), 3\sin(u), v\rangle \) for \( 0 \leq u \leq 2\pi \), \( 0 \leq v \leq 4 \).
Problem Statement |
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Determine the surface area of the cylinder given by \( \vec{r} = \langle 3\cos(u), 3\sin(u), v\rangle \) for \( 0 \leq u \leq 2\pi \), \( 0 \leq v \leq 4 \).
Final Answer |
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\( 24\pi \) |
Problem Statement |
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Determine the surface area of the cylinder given by \( \vec{r} = \langle 3\cos(u), 3\sin(u), v\rangle \) for \( 0 \leq u \leq 2\pi \), \( 0 \leq v \leq 4 \).
Solution |
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video by MIP4U
Final Answer |
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\( 24\pi \) |
close solution |
Determine the surface area of the sphere given by \( \vec{r} = \langle 2\sin(u)\cos(v), 2\sin(u)\sin(v), 2\cos(u)\rangle \) for \( 0 \leq u \leq \pi \), \( 0 \leq v \leq 2\pi \).
Problem Statement |
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Determine the surface area of the sphere given by \( \vec{r} = \langle 2\sin(u)\cos(v), 2\sin(u)\sin(v), 2\cos(u)\rangle \) for \( 0 \leq u \leq \pi \), \( 0 \leq v \leq 2\pi \).
Final Answer |
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\( 16\pi \) |
Problem Statement |
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Determine the surface area of the sphere given by \( \vec{r} = \langle 2\sin(u)\cos(v), 2\sin(u)\sin(v), 2\cos(u)\rangle \) for \( 0 \leq u \leq \pi \), \( 0 \leq v \leq 2\pi \).
Solution |
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video by MIP4U
Final Answer |
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\( 16\pi \) |
close solution |
Intermediate Problems |
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Calculate the flux of \( \vec{F} = z\hat{i} + yz\hat{j} + 2x\hat{k} \) across the upper hemisphere of the unit sphere oriented with outward-pointing normals.
Problem Statement |
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Calculate the flux of \( \vec{F} = z\hat{i} + yz\hat{j} + 2x\hat{k} \) across the upper hemisphere of the unit sphere oriented with outward-pointing normals.
Final Answer |
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\( \pi/4 \) |
Problem Statement |
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Calculate the flux of \( \vec{F} = z\hat{i} + yz\hat{j} + 2x\hat{k} \) across the upper hemisphere of the unit sphere oriented with outward-pointing normals.
Solution |
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video by Dr Chris Tisdell
Final Answer |
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\( \pi/4 \) |
close solution |