Chapter 15 Curves, Surfaces and Volumes Alberto J Chapter 15 Curves, Surfaces and Volumes Alberto J. Benavides, Adewale Awoniyi, Xiaohong Cui Department of Chemical Engineering Texas A&M University, College Station, TX

Agenda Surfaces 15.4 Surface Integrals 15.5 Volume Integrals 15.6

15.4. Surfaces Alberto J. Benavides, Adewale Awoniyi, Xiaohong Cui Department of Chemical Engineering Texas A&M University, College Station, TX

15.4.1 Parametric representation A parametric surface is a surface in the Euclidean space R3 which is defined by a parametric equation with two parameters, u and v. That is: x = x(u,v), y = y(u,v), z = z(u,v) or r = x(u,v)i + y(u,v)j + z(u,v)k Example 1: Sphere x = sin(v)cos(u) y = sin(v)sin(u) z = cos(v) (0,π) Source: http://www.math.uri.edu/~bkaskosz/flashmo/parsur/ t=v, s=u Source: http://www.math.uri.edu/~bkaskosz/flashmo/parsur/ (0,0) (2π,0) 4/26/2017

15.4.1 Parametric representation A parametric surface is a surface in the Euclidean space R3 which is defined by a parametric equation with two parameters, u and v. That is: x = x(u,v), y = y(u,v), z = z(u,v) or R = x(u,v)i + y(u,v)j + z(u,v)k Example 2: Cone x = vcos(u) y = vsin(u) z = v (0,1) http://www.math.uri.edu/~bkaskosz/flashmo/parsur/ t=v, s=u Source: http://www.math.uri.edu/~bkaskosz/flashmo/parsur/ (0,-1) (2π,-1) 4/26/2017

15.4.1 Parametric representation A parametric surface is a surface in the Euclidean space R3 which is defined by a parametric equation with two parameters, u and v. That is: x = x(u,v), y = y(u,v), z = z(u,v) or R = x(u,v)i + y(u,v)j + z(u,v)k Example 3: Mobius Band x = 2cos(u)+vcos(u/2) y = 2sin(u)+vcos(u/2) z = vsin(u/2) (0,0.5) http://www.math.uri.edu/~bkaskosz/flashmo/parsur/ t=v, s=u Source: http://www.math.uri.edu/~bkaskosz/flashmo/parsur/ (0,-0.5) (2π,-0.5) 4/26/2017

15.4.1 Parametric representation A parametric surface is a surface in the Euclidean space R3 which is defined by a parametric equation with two parameters, u and v. That is: x = x(u,v), y = y(u,v), z = z(u,v) or R = x(u,v)i + y(u,v)j + z(u,v)k Example 4: “Snake” x = (1-u)(3+cosv)cos(2πu) y = (1-u)(3+cosv)sin(2πu) z = 6u+(1-u)sin(v) (0,1) http://www.math.uri.edu/~bkaskosz/flashmo/parsur/ t=v, s=u Source: http://www.math.uri.edu/~bkaskosz/flashmo/parsur/ (0,0) (2π,0) 4/26/2017

15.4.1 Parametric representation A parametric surface is a surface in the Euclidean space R3 which is defined by a parametric equation with two parameters, u and v. That is: x = x(u,v), y = y(u,v), z = z(u,v) or R = x(u,v)i + y(u,v)j + z(u,v)k Example 5: “Shell” x = (4/3)^u*sin(v)*sin(v)*cos(u) y = (4/3)^u*sin(v)*sin(v)*sin(u) z = (4/3)^u*sin(v)*cos(v) (0,π) http://www.math.uri.edu/~bkaskosz/flashmo/parsur/ t=v, s=u Source: http://www.math.uri.edu/~bkaskosz/flashmo/parsur/ (-6,0) (1.1π,0) 4/26/2017

15.4.2 Tangent plane and normal Consider the parametric surface S specified by r. If we keep u constant by setting u = u0, then r(u0,v) moves along a curve C that lies on S as v varies. The curve C is referred to as a grid curve of constant u. Similarly, by keeping v constant, we obtain grid curves of constant v that lie on S as u varies. Source: Advanced Engineering Mathematics (2nd Edition), Michael Greenberg, Prentice Hall 4/26/2017

15.4.2 Tangent plane and normal Let p = (x(u0,v0); y(u0,v0); z(u0,v0)) be a point on S. Let C1 and C2 be the grid curves parametrized by r(u0,v) and r(u,v0) respectively. Then the derivative of r(u0,v) at the point P gives rise to a tangent vector of C1 at P, i.e., the vector: Likewise, is a tangent vector to C2 at p. In particular, the vectors ru(u0,v0) and rv(u0,v0) are contained in the tangent plane to the surface S at the point p, i.e., the plane that contains all the tangent vectors to curves lying in S and passing through p. Consequently, if ru(u0,v0) and rv(u0,v0) are not parallel, then ru(u0,v0) rv(u0,v0) is a normal vector to the tangent plane at p, from which it follows that an equation for the tangent plane is Adapted from: http://www.math.osu.edu/~khkwa/254au10/16.6withScannedExamples.pdf 4/26/2017

15.4.2 Tangent plane and normal   Or, Where d = axp + byp + czp.   Adapted from: http://www.math.osu.edu/~khkwa/254au10/16.6withScannedExamples.pdf 4/26/2017

15.4.2 Tangent plane and normal Example 6. Find an equation of the tangent plane to the given parametric surface at the specified point. Solution: Compute the tangent vectors. The cross product of these two will give us the normal. At the point ( 2, 3, 0 ), the parametric equations are: Solving this system of equations gives us u = 1, v = 1. The normal vector at point ( 2, 3, 0 ) is <-6,2,-6>. Therefore, the tangent plane at point ( 2, 3, 0 ) is: Source: http://www.math.ucla.edu/~ronmiech/Calculus_Problems/32B/chap14/section6/937d15/937_15.html 4/26/2017

15.4.2 Tangent plane and normal   4/26/2017

15.4.2 Tangent plane and normal   Adapted from: planetmath.org/encyclopedia/TangentPlane.html 4/26/2017

15.4.2 Tangent plane and normal   Adapted from: planetmath.org/encyclopedia/TangentPlane.html 4/26/2017

15. 5. Surface Integrals Alberto J 15.5. Surface Integrals Alberto J. Benavides, Adewale Awoniyi, Xiaohong Cui Department of Chemical Engineering Texas A&M University, College Station, TX

15.5 Surface Integrals – Motivation & Outline - Why Surface Integrals Are Important Applications of surface integrals include: (1) Calculating surface area (2) Calculating flux (rate of fluid flow across a surface area, etc) - Outline of This Section (1) Area element dA (2) Surface integrals 4/26/2017

15.5.1 Area Element dA Consider a surface S given parametrically by R(u,v) = x(u,v)i + y(u,v)j + z(u,v)k where R(u,v) is C1 and Ru × Rv ≠ 0 on S. Such a surface is said to be smooth. The two vectors along u = constant and v = constant curves through P, dR = Rvdv and dR = Rudu respectively, define a parallelogram (called area element on S) lying in the tangent plane to S at P. According to the geometrical significance of the cross product of two vectors, the area of the parallelogram is dA = ║Rudu × Rvdv║ = ║Ru × Rv║dudv Figure 1. Area element dA (Adapted from: http://en.wikipedia.org/wiki/ File:Surface_integral1.svg) dA P dR = Rudu v = constant dR = Rvdv u = constant Adapted from: Advanced Engineering Mathematics (2nd Edition), Michael Greenberg, Prentice Hall 4/26/2017

dA = ║Rudu × Rvdv║ = ║Ru × Rv║dudv 15.5.1 Area Element dA dA = ║Rudu × Rvdv║ = ║Ru × Rv║dudv To obtain a computational version of the above expression, cross Ru = xui + yuj + zuk with Rv = xvi + yvj + zvk. The norm of the resulting vector is the square root of the sum of the square of its components. Thus we obtain Figure 1. Area element dA (Adapted from: http://en.wikipedia.org/wiki/ File:Surface_integral1.svg) dA P dR = Rudu v = constant dR = Rvdv u = constant   Adapted from: Advanced Engineering Mathematics (2nd Edition), Michael Greenberg, Prentice Hall 4/26/2017

15.5.1 Area Element dA 4/26/2017 Adapted from:     Adapted from: Advanced Engineering Mathematics (2nd Edition), Michael Greenberg, Prentice Hall 4/26/2017

Figure 2. dA in polar coordinates 15.5.1 Area Element dA     Figure 2. dA in polar coordinates (Source: Advanced Engineering Mathematics (2nd Edition), Michael Greenberg, Prentice Hall)   Adapted from: Advanced Engineering Mathematics (2nd Edition), Michael Greenberg, Prentice Hall 4/26/2017

15.5.1 Area Element dA 4/26/2017 Adapted from:     Adapted from: Advanced Engineering Mathematics (2nd Edition), Michael Greenberg, Prentice Hall 4/26/2017

dA = ║Rudu × Rvdv║ = ║Ru × Rv║dudv 15.5.1 Area Element dA Calculating Surface Area According to we define the area of the curved surface S by the double integral Figure 1. Area element dA (Adapted from: http://en.wikipedia.org/wiki/ File:Surface_integral1.svg) dA P dR = Rudu v = constant dR = Rvdv u = constant dA = ║Rudu × Rvdv║ = ║Ru × Rv║dudv   Adapted from: Advanced Engineering Mathematics (2nd Edition), Michael Greenberg, Prentice Hall 4/26/2017

15.5.1 Area Element dA       http://tutorial.math.lamar.edu/Classes/CalcIII/SurfaceArea.aspx     Source: http://tutorial.math.lamar.edu/Classes/CalcIII/SurfaceArea.aspx 4/26/2017

15.5.2 Surface Integrals         http://en.wikipedia.org/wiki/File:Surface_integral_illustration.png   Adapted from: Advanced Engineering Mathematics (2nd Edition), Michael Greenberg, Prentice Hall 4/26/2017

15.5.2 Surface Integrals Calculation (1) Surface is parametrized in the form R(u,v)     http://en.wikipedia.org/wiki/File:Surface_integral_illustration.png 4/26/2017

15.5.2 Surface Integrals     http://tutorial.math.lamar.edu/Classes/CalcIII/SurfaceIntegrals.aspx   Source: http://tutorial.math.lamar.edu/Classes/CalcIII/SurfaceIntegrals.aspx 4/26/2017

15.5.2 Surface Integrals     http://tutorial.math.lamar.edu/Classes/CalcIII/SurfaceIntegrals.aspx   Source: http://tutorial.math.lamar.edu/Classes/CalcIII/SurfaceIntegrals.aspx 4/26/2017

15.5.2 Surface Integrals     http://www.math.oregonstate.edu/home/programs/undergrad/CalculusQuestStudyGuides/vcalc/flux/flux.html   Source: http://www.math.oregonstate.edu/home/programs/undergrad/CalculusQuestStudyGuides/vcalc/flux/flux.html 4/26/2017

15.5.2 Surface Integrals     http://www.math.oregonstate.edu/home/programs/undergrad/CalculusQuestStudyGuides/vcalc/flux/flux.html Source: http://www.math.oregonstate.edu/home/programs/undergrad/CalculusQuestStudyGuides/vcalc/flux/flux.html 4/26/2017

15. 6. Volume Integrals Alberto J 15.6. Volume Integrals Alberto J. Benavides, Adewale Awoniyi, Xiaohong Cui Department of Chemical Engineering Texas A&M University, College Station, TX

15.6 Volume Integrals – Motivation & Outline - Why Volume Integrals Are Important Applications of volume integrals include: (1) Calculating volume of a given region (2) Physical applications: calculating the moment of inertia, gravitational force, etc (3) Solving partial differential equations - Outline of This Section (1) Volume element dV (2) Volume integrals 4/26/2017

15.6.1 Volume Element dV Consider the position vector R given parametrically by R(u,v,w) = x(u,v,w)i + y(u,v,w)j + z(u,v,w)k where R(u,v,w) is C1 and Ru, Rv and Rw are linearly independent. For each fixed w, the parametrization defines a surface. As we vary w we produce a family of such surfaces which will generate a volume. The three vectors dR = Rudu, dR = Rvdv and dR = Rwdw determine a parallelepiped of nonzero volume (called volume element). Figure 1. Volume element dV (Source: Advanced Engineering Mathematics (2nd Edition), Michael Greenberg, Prentice Hall) Adapted from: Advanced Engineering Mathematics (2nd Edition), Michael Greenberg, Prentice Hall 4/26/2017

Figure 1. Volume element dV   Figure 1. Volume element dV (Source: Advanced Engineering Mathematics (2nd Edition), Michael Greenberg, Prentice Hall)   Adapted from: Advanced Engineering Mathematics (2nd Edition), Michael Greenberg, Prentice Hall 4/26/2017

15.6.1 Volume Element dV   Figure 2. Volume element in cylindrical coordinates (Source:http://keep2.sjfc.edu/faculty/kgreen/vector/Block3/jacob/node9.html)   Adapted from: Advanced Engineering Mathematics (2nd Edition), Michael Greenberg, Prentice Hall 4/26/2017

15.6.1 Volume Element dV Two special cases of calculating dV (1) Cyclindrical Coordinates Example 1. (continue) Summary of information obtained from the position vector in cylindrical coordinates is as follows. Source: Advanced Engineering Mathematics (2nd Edition), Michael Greenberg, Prentice Hall 4/26/2017

15.6.1 Volume Element dV   Figure 3. Volume element in spherical coordinates (Source:http://keep2.sjfc.edu/faculty/kgreen/vector/Block3/jacob/node9.html) Adapted from: Advanced Engineering Mathematics (2nd Edition), Michael Greenberg, Prentice Hall 4/26/2017

15.6.1 Volume Element dV   Figure 3. Volume element in spherical coordinates (Source: Salas, Hille, Etgen Calculus: One and Several Variables, John Wiley & Sons, Inc)   Adapted from: Advanced Engineering Mathematics (2nd Edition), Michael Greenberg, Prentice Hall 4/26/2017

15.6.1 Volume Element dV Two special cases of calculating dV (2) Spherical Coordinates Example 2. (continue) Summary of information obtained from the position vector in spherical coordinates is as follows. Source: Advanced Engineering Mathematics (2nd Edition), Michael Greenberg, Prentice Hall 4/26/2017

Figure 1. Volume element dV Calculating Volume of a Given Region According to we define the volume of a region V by the triple integral     Figure 1. Volume element dV (Source: Advanced Engineering Mathematics (2nd Edition), Michael Greenberg, Prentice Hall) Adapted from: Advanced Engineering Mathematics (2nd Edition), Michael Greenberg, Prentice Hall 4/26/2017

Figure 4. The region described in Example 3 15.6.1 Volume Element dV   http://tutorial.math.lamar.edu/Classes/CalcIII/TripleIntegrals.aspx Figure 4. The region described in Example 3 (Source: http://tutorial.math.lamar.edu/Classes/CalcIII/TripleIntegrals.aspx/) Source: http://tutorial.math.lamar.edu/Classes/CalcIII/TripleIntegrals.aspx 4/26/2017

Figure 4. The region described in Example 3 15.6.1 Volume Element dV   Figure 4. The region described in Example 3 (Source: http://tutorial.math.lamar.edu/Classes /CalcIII/TripleIntegrals.aspx/) Solution: http://tutorial.math.lamar.edu/Classes/CalcIII/TripleIntegrals.aspx Source: http://tutorial.math.lamar.edu/Classes/CalcIII/TripleIntegrals.aspx 4/26/2017

15.6.2 Volume Integrals Definition We define volume integral of a given function f over a given region V in 3-space for if x, y, z are parametrized as u, v, w, then   Where R is the region in u, v, w space corresponding to the region V in the x, y, z space. http://en.wikipedia.org/wiki/File:Surface_integral_illustration.png Adapted from: Advanced Engineering Mathematics (2nd Edition), Michael Greenberg, Prentice Hall 4/26/2017

15.6.2 Volume Integrals     http://tutorial.math.lamar.edu/Classes/CalcIII/TripleIntegrals.aspx   Source: http://tutorial.math.lamar.edu/Classes/CalcIII/TripleIntegrals.aspx 4/26/2017

15.6.2 Volume Integrals       http://tutorial.math.lamar.edu/Classes/CalcIII/TripleIntegrals.aspx Source: http://tutorial.math.lamar.edu/Classes/CalcIII/TripleIntegrals.aspx 4/26/2017

15.6.2 Volume Integrals     http://tutorial.math.lamar.edu/Classes/CalcIII/TripleIntegrals.aspx   Source: http://tutorial.math.lamar.edu/Classes/CalcIII/TripleIntegrals.aspx 4/26/2017

15.6.2 Volume Integrals     http://spiff.rit.edu/classes/phys350/phys_iii.html/ Figure 6. The hollow sphere described in Example 5 (Source: http://spiff.rit.edu/classes/phy s350/phys_iii.html/) Adapted from: Advanced Engineering Mathematics (2nd Edition), Michael Greenberg, Prentice Hall 4/26/2017

15.6.2 Volume Integrals   Figure 6. The hollow sphere described in Example 5 (Source: http://spiff.rit.edu/classes/phy s350/phys_iii.html/) Adapted from: Advanced Engineering Mathematics (2nd Edition), Michael Greenberg, Prentice Hall 4/26/2017

15.6.2 Volume Integrals   Figure 6. The hollow sphere described in Example 5 (Source: http://spiff.rit.edu/classes/phy s350/phys_iii.html/) Adapted from: Advanced Engineering Mathematics (2nd Edition), Michael Greenberg, Prentice Hall 4/26/2017