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Show that the surface area of cylinder \(x^2 + y^2 = r^2, \, 0 \leq z \leq h\) is \(2\pi rh\). Example \(\PageIndex{10}\): Calculating the Surface Integral of a Piece of a Sphere. Informally, the surface integral of a scalar-valued function is an analog of a scalar line integral in one higher dimension. Show that the surface area of the sphere \(x^2 + y^2 + z^2 = r^2\) is \(4 \pi r^2\). A “simple” surface-integral over the unit-sphere [closed] Ask Question Asked 5 days ago. Closed surfaces such as spheres are orientable: if we choose the outward normal vector at each point on the surface of the sphere, then the unit normal vectors vary continuously. By the definition of the line integral (Section 16.2), \[\begin{align*} m &= \iint_S x^2 yz \, dS \\[4pt] A flat sheet of metal has the shape of surface \(z = 1 + x + 2y\) that lies above rectangle \(0 \leq x \leq 4\) and \(0 \leq y \leq 2\). The methods discussed in the present paper are not optimal, but they are well-suited to the solution of integral equations. To get an idea of the shape of the surface, we first plot some points. &= - 55 \int_0^{2\pi} \int_0^1 \langle 8v \, \cos u, \, 8v \, \sin u, \, v^2\rangle \cdot \langle 0, 0, -v \rangle\, \, dv \,du\\[4pt] We also could choose the inward normal vector at each point to give an “inward” orientation, which is the negative orientation of the surface. 1 Lecture 35 : Surface Area; Surface Integrals In the previous lecture we deflned the surface area a(S) of the parametric surface S, deflned by r(u;v) on T, by the double integral a(S) = RR T k ru £rv k dudv: (1) We will now drive a formula for the area of a surface deflned by the graph of a function. Then the curve traced out by the parameterization is \(\langle \cos u, \, \sin u, \, K \rangle \), which gives a circle in plane \(z = K\) with radius 1 and center \((0, 0, K)\). So, for our example we will have. How could we calculate the mass flux of the fluid across \(S\)? A piece of metal has a shape that is modeled by paraboloid \(z = x^2 + y^2, \, 0 \leq z \leq 4,\) and the density of the metal is given by \(\rho (x,y,z) = z + 1\). Let \(S\) be a smooth orientable surface with parameterization \(\vecs r(u,v)\). If we want to find the flow rate (measured in volume per time) instead, we can use flux integral. After that the integral is a standard double integral and by this point we should be able to deal with that. Note that \(\vecs t_u = \langle 1, 2u, 0 \rangle\) and \(\vecs t_v = \langle 0,0,1 \rangle\). A surface integral over a vector field is also called a flux integral. In analytic geometry, a sphere with center (x0, y0, z0) and radius r is the locus of all points (x, y, z) such that Credits. The surface integral of the vector field \(\mathbf{F}\) over the oriented surface \(S\) (or the flux of the vector field \(\mathbf{F}\) across the surface \(S\)) can be written in one of the following forms: Equation \ref{scalar surface integrals} allows us to calculate a surface integral by transforming it into a double integral. In this case we don’t need to do any parameterization since it is set up to use the formula that we gave at the start of this section. In other words, the top of the cylinder will be at an angle. To obtain a parameterization, let \(\alpha\) be the angle that is swept out by starting at the positive z-axis and ending at the cone, and let \(k = \tan \alpha\). Example \(\PageIndex{2}\): Describing a Surface. Calculate the lateral surface area (the area of the “side,” not including the base) of the right circular cone with height h and radius r. Before calculating the surface area of this cone using Equation \ref{equation1}, we need a parameterization. The surface in Figure \(\PageIndex{8a}\) can be parameterized by, \[\vecs r(u,v) = \langle (2 + \cos v) \cos u, \, (2 + \cos v) \sin u, \, \sin v \rangle, \, 0 \leq u < 2\pi, \, 0 \leq v < 2\pi \nonumber\], (we can use technology to verify). A cast-iron solid cylinder is given by inequalities \(x^2 + y^2 \leq 1, \, 1 \leq z \leq 4\). The image of this parameterization is simply point \((1,2)\), which is not a curve. Note that we can form a grid with lines that are parallel to the \(u\)-axis and the \(v\)-axis in the \(uv\)-plane. Therefore, the mass flow rate is \(7200\pi \, \text{kg/sec/m}^2\). Then, \(S\) can be parameterized with parameters \(x\) and \(\theta\) by, \[\vecs r(x, \theta) = \langle x, f(x) \, \cos \theta, \, f(x) \sin \theta \rangle, \, a \leq x \leq b, \, 0 \leq x \leq 2\pi.\], Example \(\PageIndex{7}\): Calculating Surface Area. &= 80 \int_0^{2\pi} \int_0^{\pi/2} 54 \, \sin^3 \phi + 27 \, \cos^2 \phi \, \sin \phi \, d\phi \, d\theta \\ The second method for evaluating a surface integral is for those surfaces that are given by the parameterization. For scalar surface integrals, we chop the domain region (no longer a curve) into tiny pieces and proceed in the same fashion. In this sense, surface integrals expand on our study of line integrals. This is not an issue though, because Equation \ref{scalar surface integrals} does not place any restrictions on the shape of the parameter domain. Use the standard parameterization of a cylinder and follow the previous example. First, we calculate \(\displaystyle \iint_{S_1} z^2 \,dS.\) To calculate this integral we need a parameterization of \(S_1\). I The area of a surface in space. In addition to modeling fluid flow, surface integrals can be used to model heat flow. If you imagine placing a normal vector at a point on the strip and having the vector travel all the way around the band, then (because of the half-twist) the vector points in the opposite direction when it gets back to its original position. &= \langle 4 \, \cos \theta \, \sin^2 \phi, \, 4 \, \sin \theta \, \sin^2 \phi, \, 4 \, \cos^2 \theta \, \cos \phi \, \sin \phi + 4 \, \sin^2 \theta \, \cos \phi \, \sin \phi \rangle \\[4 pt] To find the heat flow, we need to calculate flux integral \[\iint_S -k\vecs \nabla T \cdot dS.\] Notice that \(S\) is not a smooth surface but is piecewise smooth, since \(S\) is the union of three smooth surfaces (the circular top and bottom, and the cylindrical side). We can extend the concept of a line integral to a surface integral to allow us to perform this integration. One line is given by \(x = u_i, \, y = v\); the other is given by \(x = u, \, y = v_j\). The surface integral will have a \(dS\) while the standard double integral will have a \(dA\). Now at this point we can proceed in one of two ways. Some surfaces are twisted in such a fashion that there is no well-defined notion of an “inner” or “outer” side. \(r \, \cos \theta \, \sin \phi, \, r \, \sin \theta \, \sin \phi, \, r \, \cos \phi \rangle, \, 0 \leq \theta < 2\pi, \, 0 \leq \phi \leq \pi.\), \(\vecs t_{\theta} = \langle -r \, \sin \theta \, \sin \phi, \, r \, \cos \theta \, \sin \phi, \, 0 \rangle\), \(\vecs t_{\phi} = \langle r \, \cos \theta \, \cos \phi, \, r \, \sin \theta \, \cos \phi, \, -r \, \sin \phi \rangle.\), \[ \begin{align*}\vecs t_{\phi} \times \vecs t_{\theta} &= \langle r^2 \cos \theta \, \sin^2 \phi, \, r^2 \sin \theta \, \sin^2 \phi, \, r^2 \sin^2 \theta \, \sin \phi \, \cos \phi + r^2 \cos^2 \theta \, \sin \phi \, \cos \phi \rangle \\[4pt] &= \langle r^2 \cos \theta \, \sin^2 \phi, \, r^2 \sin \theta \, \sin^2 \phi, \, r^2 \sin \phi \, \cos \phi \rangle. If a region R is not flat, then it is called a surface as shown in the illustration. Thus, a surface integral is similar to a line integral but in one higher dimension. The definition of a surface integral of a vector field proceeds in the same fashion, except now we chop surface \(S\) into small pieces, choose a point in the small (two-dimensional) piece, and calculate \(\vecs{F} \cdot \vecs{N}\) at the point. Similarly, points \(\vecs r(\pi, 2) = (-1,0,2)\) and \(\vecs r \left(\dfrac{\pi}{2}, 4\right) = (0,1,4)\) are on \(S\). \end{align*}\], \[\begin{align*} \iint_{S_2} z \, dS &= \int_0^{\pi/6} \int_0^{2\pi} f (\vecs r(\phi, \theta))||\vecs t_{\phi} \times \vecs t_{\theta}|| \, d\theta \, d\phi \\ Practice computing a surface integral over a sphere. The definition of a scalar line integral can be extended to parameter domains that are not rectangles by using the same logic used earlier. Sort by: Top Voted. To visualize \(S\), we visualize two families of curves that lie on \(S\). Now we need \({\vec r_z} \times {\vec r_\theta }\). To calculate the surface integral, we first need a parameterization of the cylinder. Suppose that \(u\) is a constant \(K\). \end{align*}\], \[ \begin{align*} \pi k h^2 \sqrt{1 + k^2} &= \pi \dfrac{r}{h}h^2 \sqrt{1 + \dfrac{r^2}{h^2}} \\[4pt] &= \pi r h \sqrt{1 + \dfrac{r^2}{h^2}} \\[4pt] \\[4pt] &= \pi r \sqrt{h^2 + h^2 \left(\dfrac{r^2}{h^2}\right) } \\[4pt] &= \pi r \sqrt{h^2 + r^2}. 1. &= - 55 \int_0^{2\pi} \int_1^4 \langle 2v \, \cos u, \, 2v \, \sin u, \, \cos^2 u + \sin^2 u \rangle \cdot \langle \cos u, \, \sin u, \, 0 \rangle \, dv\, du \\[4pt] The tangent vectors are \(\vecs t_u = \langle - kv \, \sin u, \, kv \, \cos u, \, 0 \rangle\) and \(\vecs t_v = \langle k \, \cos u, \, k \, \sin u, \, 1 \rangle\). Since we are working on the upper half of the sphere here are the limits on the parameters. The rate of heat flow across surface S in the object is given by the flux integral, \[\iint_S \vecs F \cdot dS = \iint_S -k \vecs \nabla T \cdot dS.\], Example \(\PageIndex{15}\): Calculating Heat Flow. \end{align*}\], By Equation \ref{equation1}, the surface area of the cone is, \[ \begin{align*}\iint_D ||\vecs t_u \times \vecs t_v|| \, dA &= \int_0^h \int_0^{2\pi} kv \sqrt{1 + k^2} \,du\, dv \\[4pt] &= 2\pi k \sqrt{1 + k^2} \int_0^h v \,dv \\[4pt] &= 2 \pi k \sqrt{1 + k^2} \left[\dfrac{v^2}{2}\right]_0^h \\[4pt] \\[4pt] &= \pi k h^2 \sqrt{1 + k^2}. \nonumber\], From the material we have already studied, we know that, \[\Delta S_{ij} \approx ||\vecs t_u (P_{ij}) \times \vecs t_v (P_{ij})|| \,\Delta u \,\Delta v. \nonumber\], \[\iint_S f(x,y,z) \,dS \approx \lim_{m,n\rightarrow\infty} \sum_{i=1}^m \sum_{j=1}^n f(P_{ij})|| \vecs t_u(P_{ij}) \times \vecs t_v(P_{ij}) ||\,\Delta u \,\Delta v. \nonumber\]. \end{align*}\]. With the standard parameterization of a cylinder, Equation \ref{equation1} shows that the surface area is \(2 \pi rh\). This surface is a disk in plane \(z = 1\) centered at \((0,0,1)\). Here are the ranges for \(y\) and \(z\). Let \(\vecs{F}\) be a continuous vector field with a domain that contains oriented surface \(S\) with unit normal vector \(\vecs{N}\). With a parameterization in hand, we can calculate the surface area of the cone using Equation \ref{equation1}. By Equation \ref{scalar surface integrals}, \[\begin{align*} \iint_S 5 \, dS &= 5 \iint_D u \sqrt{1 + 4u^2} \, dA \\ A surface parameterization \(\vecs r(u,v) = \langle x(u,v), y(u,v), z(u,v) \rangle\) is smooth if vector \(\vecs r_u \times \vecs r_v\) is not zero for any choice of \(u\) and \(v\) in the parameter domain. Surface area and surface integrals. The unit vector points outwards from a closed surface and is usually denoted by ˆn. The second step is to define the surface area of a parametric surface. Then, the unit normal vector is given by \(\vecs N = \dfrac{\vecs t_u \times \vecs t_v}{||\vecs t_u \times \vecs t_v||}\) and, from Equation \ref{surfaceI}, we have, \[\begin{align*} \int_C \vecs F \cdot \vecs N\, dS &= \iint_S \vecs F \cdot \dfrac{\vecs t_u \times \vecs t_v}{||\vecs t_u \times \vecs t_v||} \,dS \\[4pt] &= \sqrt{6} \int_0^4 \dfrac{22x^2}{3} + 2x^3 \,dx \\[4pt] Given that the thermal conductivity of cast iron is 55, find the heat flow across the boundary of the solid if this boundary is oriented outward. However, if we wish to integrate over a surface (a two-dimensional object) rather than a path (a one-dimensional object) in space, then we need a new kind of integral that can handle integration over objects in higher dimensions. So it could be defined by x squared plus y squared plus z squared is equal to 1. Before calculating any integrals, note that the gradient of the temperature is \(\vecs \nabla T = \langle 2xz, \, 2yz, \, x^2 + y^2 \rangle\). This approximation becomes arbitrarily close to \(\displaystyle \lim_{m,n\rightarrow\infty} \sum_{i=1}^m \sum_{j=1}^n f(P_{ij}) \Delta S_{ij}\) as we increase the number of pieces \(S_{ij}\) by letting \(m\) and \(n\) go to infinity. A plot of S is given below. For now, assume the parameter domain \(D\) is a rectangle, but we can extend the basic logic of how we proceed to any parameter domain (the choice of a rectangle is simply to make the notation more manageable). Parameterization \(\vecs r(u,v) = \langle x(u,v), y(u,v), z(u,v) \rangle\) is a regular parameterization if \(\vecs r_u \times \vecs r_v\) is not zero for point \((u,v)\) in the parameter domain. we can always use this form for these kinds of surfaces as well. Is the surface parameterization \(\vecs r(u,v) = \langle u^{2v}, v + 1, \, \sin u \rangle, \, 0 \leq u \leq 2, \, 0 \leq v \leq 3\) smooth? To compute the flow rate of the fluid in Example, we simply remove the density constant, which gives a flow rate of \(90 \pi \, m^3/sec\). step is a bit of notation – when the surface is closed, we often put a ring around the integral: Every bit in our surface Φ= ⋅∫ EnˆdA Our steps in calculating it are always going to be 1. Since this is for 1/8th of a sphere, the total surface integral of the sphere is 4 π a 2. Describe the surface with parameterization, \[\vecs{r} (u,v) = \langle 2 \, \cos u, \, 2 \, \sin u, \, v \rangle, \, 0 \leq u \leq 2\pi, \, -\infty < v < \infty \nonumber\]. Notice that vectors, \[\vecs r_u = \langle - (2 + \cos v)\sin u, \, (2 + \cos v) \cos u, 0 \rangle \nonumber\], \[\vecs r_v = \langle -\sin v \, \cos u, \, - \sin v \, \sin u, \, \cos v \rangle \nonumber\], exist for any choice of \(u\) and \(v\) in the parameter domain, and, \[ \begin{align*} \vecs r_u \times \vecs r_v &= \begin{vmatrix} \mathbf{\hat{i}}& \mathbf{\hat{j}}& \mathbf{\hat{k}} \\ -(2 + \cos v)\sin u & (2 + \cos v)\cos u & 0\\ -\sin v \, \cos u & - \sin v \, \sin u & \cos v \end{vmatrix} \\[4pt] &= [(2 + \cos v)\cos u \, \cos v] \mathbf{\hat{i}} + [2 + \cos v) \sin u \, \cos v] \mathbf{\hat{j}} + [(2 + \cos v)\sin v \, \sin^2 u + (2 + \cos v) \sin v \, \cos^2 u]\mathbf{\hat{k}} \\[4pt] &= [(2 + \cos v)\cos u \, \cos v] \mathbf{\hat{i}} + [(2 + \cos v) \sin u \, \cos v]\mathbf{\hat{j}} + [(2 + \cos v)\sin v ] \mathbf{\hat{k}}. If you think of the normal field as describing water flow, then the side of the surface that water flows toward is the “negative” side and the side of the surface at which the water flows away is the “positive” side. This is easy enough to do. The fact that the derivative is the zero vector indicates we are not actually looking at a curve. What we are doing now is the analog of this in space. Consider the parameter domain for this surface. A parameterized surface is given by a description of the form, \[\vecs{r}(u,v) = \langle x (u,v), \, y(u,v), \, z(u,v)\rangle.\]. Therefore, the lateral surface area of the cone is \(\pi r \sqrt{h^2 + r^2}\). The surface integral of the (continuous) function f(x,y,z) over the surface S is denoted by (1) Z Z S f(x,y,z)dS . \nonumber\]. If we only care about a piece of the graph of \(f\) - say, the piece of the graph over rectangle \([ 1,3] \times [2,5]\) - then we can restrict the parameter domain to give this piece of the surface: \[\vecs r(x,y) = \langle x,y,x^2y \rangle, \, 1 \leq x \leq 3, \, 2 \leq y \leq 5.\]. Example \(\PageIndex{14}\):Calculating Mass Flow Rate. Although this parameterization appears to be the parameterization of a surface, notice that the image is actually a line (Figure \(\PageIndex{7}\)). Therefore, a parameterization of this cone is, \[\vecs s(u,v) = \langle kv \, \cos u, \, kv \, \sin u, \, v \rangle, \, 0 \leq u < 2\pi, \, 0 \leq v \leq h. \nonumber\]. \end{align*}\], Therefore, the rate of heat flow across \(S\) is, \[\dfrac{55\pi}{2} - \dfrac{55\pi}{2} - 110\pi = -110\pi. Then, \[\vecs t_u \times \vecs t_v = \begin{vmatrix} \mathbf{\hat i} & \mathbf{\hat j} & \mathbf{\hat k} \\ -\sin u & \cos u & 0 \\ 0 & 0 & 1 \end{vmatrix} = \langle \cos u, \, \sin u, \, 0 \rangle\]. Recall that curve parameterization \(\vecs r(t), \, a \leq t \leq b\) is smooth if \(\vecs r'(t)\) is continuous and \(\vecs r'(t) \neq \vecs 0\) for all \(t\) in \([a,b]\). Given the radius r of the sphere, the total surface area is. Divide rectangle \(D\) into subrectangles \(D_{ij}\) with horizontal width \(\Delta u\) and vertical length \(\Delta v\). Integrating spheres are very versatile optical elements, which are designed to achieve homogenous distribution of optical radiation by means of multiple Lambertian reflections at the sphere's inner surface. In a similar fashion, we can use scalar surface integrals to compute the mass of a sheet given its density function. Example \(\PageIndex{3}\): Finding a Parameterization. Describe the surface integral of a vector field. The way to tell them apart is by looking at the differentials. The tangent plane at \(P_{ij}\) contains vectors \(\vecs t_u(P_{ij})\) and \(\vecs t_v(P_{ij})\) and therefore the parallelogram spanned by \(\vecs t_u(P_{ij})\) and \(\vecs t_v(P_{ij})\) is in the tangent plane. Compute the surface integral where S is that part of the plane x+y+z=2 in the first octant. Notice that if we change the parameter domain, we could get a different surface. Recall that curve parameterization \(\vecs r(t), \, a \leq t \leq b\) is regular (or smooth) if \(\vecs r'(t) \neq \vecs 0\) for all \(t\) in \([a,b]\). &= 2\pi \left[ \dfrac{1}{64} \left(2 \sqrt{4b^2 + 1} (8b^3 + b) \, \sinh^{-1} (2b) \right)\right]. To confirm this, notice that, \[\begin{align*} x^2 + y^2 &= (u \, \cos v)^2 + (u \, \sin v)^2 \\[4pt] &= u^2 \cos^2 v + u^2 sin^2 v \\[4pt] &= u^2 \\[4pt] &=z\end{align*}\]. To create a Möbius strip, take a rectangular strip of paper, give the piece of paper a half-twist, and the glue the ends together (Figure \(\PageIndex{20}\)). \nonumber\]. Find the mass flow rate of the fluid across \(S\). A surface integral is like a line integral in one higher dimension.

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