VECTOR CALCULUS

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16. VECTOR CALCULUS. VECTOR CALCULUS. So far, we have considered special types of surfaces: Cylinders Quadric surfaces Graphs of functions of two variables Level surfaces of functions of three variables. VECTOR CALCULUS.

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16

VECTOR CALCULUS

VECTOR CALCULUS
• So far, we have considered special types of surfaces:
• Cylinders
• Graphs of functions of two variables
• Level surfaces of functions of three variables
VECTOR CALCULUS
• Here, we use vector functions to describe more general surfaces, called parametric surfaces, and compute their areas.
VECTOR CALCULUS
• Then, we take the general surface area formula and see how it applies to special surfaces.

VECTOR CALCULUS

16.6

Parametric Surfaces and their Areas

• In this section, we will learn about:
• Various types of parametric surfaces
• and computing their areas using vector functions.
INTRODUCTION
• We describe a space curve by a vector function r(t) of a single parameter t.
• Similarly,we can describe a surface by a vector function r(u, v) of two parameters u and v.
INTRODUCTION

Equation 1

• We suppose that r(u, v) = x(u, v) i + y(u, v) j + z (u, v) kis a vector-valued function defined on a region D in the uv-plane.
INTRODUCTION
• So x, y, and z—the component functions of r—are functions of the two variables u and v with domain D.
PARAMETRIC SURFACE

Equations 2

• The set of all points (x, y, z) in such that x = x(u, v) y = y(u, v) z = z(u, v)and (u, v) varies throughout D, is called a parametric surface S.
• Equations 2 are called parametric equationsof S.
PARAMETRIC SURFACES
• Each choice of u and v gives a point on S.
• By making all choices, we get all of S.
PARAMETRIC SURFACES
• In other words, the surface S is traced out by the tip of the position vector r(u, v) as (u, v) moves throughout the region D.
PARAMETRIC SURFACES

Example 1

• Identify and sketch the surface with vector equation r(u, v) = 2 cos ui + vj + 2 sin uk
• The parametric equations for this surface are: x = 2 cos uy = vz = 2 sin u
PARAMETRIC SURFACES

Example 1

• So, for any point (x, y, z) on the surface, we have:x2 + z2 = 4 cos2u + 4 sin2u = 4
• This means that vertical cross-sections parallel to the xz-plane (that is, with y constant) are all circles with radius 2.
PARAMETRIC SURFACES

Example 1

• Since y = v and no restriction is placed on v, the surface is a circular cylinder with radius 2 whose axis is the y-axis.
PARAMETRIC SURFACES
• In Example 1, we placed no restrictions on the parameters u and v.
• So,we obtained the entire cylinder.
PARAMETRIC SURFACES
• If, for instance, we restrict u and v by writing the parameter domain as 0 ≤ u ≤ π/2 0 ≤ v ≤ 3then x≥ 0 z ≥ 0 0 ≤ y ≤ 3
PARAMETRIC SURFACES
• In that case, we get the quarter-cylinder with length 3.
PARAMETRIC SURFACES
• If a parametric surface S is given by a vector function r(u, v), then there are two useful families of curves that lie on S—one with u constant and the other with v constant.
• These correspond to vertical and horizontal lines in the uv-plane.
PARAMETRIC SURFACES
• Keeping u constant by putting u = u0, r(u0, v) becomes a vector function of the single parameter v and defines a curve C1 lying on S.
GRID CURVES
• Similarly, keeping v constant by putting v = v0, we get a curve C2 given by r(u, v0) that lies on S.
• We call these curves grid curves.
GRID CURVES
• In Example 1, for instance, the grid curves obtained by:
• Letting u be constant are horizontal lines.
• Letting v be constant are circles.
GRID CURVES
• In fact, when a computer graphs a parametric surface, it usually depicts the surface by plotting these grid curves—as we see in the following example.
GRID CURVES

Example 2

• Use a computer algebra system to graph the surface
• r(u, v) = <(2 + sin v) cos u, (2 + sin v) sin u, u + cos v>
• Which grid curves have u constant?
• Which have v constant?
GRID CURVES

Example 2

• We graph the portion of the surface with parameter domain 0 ≤ u ≤ 4π, 0 ≤v≤ 2π
• It has the appearance of a spiral tube.
GRID CURVES

Example 2

• To identify the grid curves, we write the corresponding parametric equations:x = (2 + sin v) cos u y = (2 + sin v) sin uz = u + cos v
GRID CURVES

Example 2

• If v is constant, then sin v and cos v are constant.
• So, the parametric equations resemble those of the helix in Example 4 in Section 13.1
GRID CURVES

Example 2

• So, the grid curves with v constant are the spiral curves.
• We deduce that the grid curves with u constant must be the curves that look like circles.
GRID CURVES

Example 2

• Further evidence for this assertion is that, if u is kept constant, u = u0, then the equation z = u0 + cos vshows that the z-values vary from u0 – 1 to u0 + 1.
PARAMETRIC REPRESENTATION
• In Examples 1 and 2 we were given a vector equation and asked to graph the corresponding parametric surface.
• In the following examples, however, we are given the more challenging problem of finding a vector function to represent a given surface.
• In the rest of the chapter, we will often need to do exactly that.
PARAMETRIC REPRESENTATIONS

Example 3

• Find a vector function that represents the plane that:
• Passes through the point P0 with position vector r0.
• Contains two nonparallel vectors a and b.
PARAMETRIC REPRESENTATIONS

Example 3

• If P is any point in the plane, we can get from P0 to P by moving a certain distance in the direction of a and another distance in the direction of b.
• So, there are scalars u and v such that: = ua + vb
PARAMETRIC REPRESENTATIONS

Example 3

• The figure illustrates how this works, by means of the Parallelogram Law, for the case where u and v are positive.
PARAMETRIC REPRESENTATIONS

Example 3

• If r is the position vector of P, then
• So, the vector equation of the plane can be written as: r(u, v) = r0 + ua + vbwhere u and v are real numbers.
PARAMETRIC REPRESENTATIONS

Example 3

• If we write r = <x, y, z>r0 = <x0, y0, z0>a = <a1, a2, a3>b = <b1, b2, b3>we can write the parametric equations of the plane through the point (x0, y0, z0) as:
• x = x0 + ua1 + vb1y = y0 + ua2 + vb2z = z0 + ua3 + vb3
PARAMETRIC REPRESENTATIONS

Example 4

• Find a parametric representation of the spherex2 + y2 + z2 = a2
• The sphere has a simple representation ρ = ain spherical coordinates.
• So, let’s choose the angles Φand θin spherical coordinates as the parameters (Section 15.8).
PARAMETRIC REPRESENTATIONS

Example 4

• Then, putting ρ = ain the equations for conversion from spherical to rectangular coordinates (Equations 1 in Section 15.8), we obtain:
• x = a sin Φ cos θy = asin Φ sin θ
• z = a cos Φ
• as the parametric equations of the sphere.
PARAMETRIC REPRESENTATIONS

Example 4

• The corresponding vector equation is: r(Φ, θ) = a sin Φ cos θi + a sin Φ sin θj + a cos Φk
• We have 0 ≤ Φ ≤ πand 0 ≤ θ≤ 2π.
• So,the parameter domain is the rectangle D = [0, π] x [0, 2π]
PARAMETRIC REPRESENTATIONS

Example 4

• The grid curves with:
• Φ constant are the circles of constant latitude (including the equator).
• θ constant are the meridians (semicircles), which connect the north and south poles.
APPLICATIONS—COMPUTER GRAPHICS
• One of the uses of parametric surfaces is in computer graphics.
COMPUTER GRAPHICS
• The figure shows the result of trying to graph the sphere x2 + y2 + z2 = 1 by:
• Solving the equation for z.
• Graphing the top and bottom hemispheres separately.
COMPUTER GRAPHICS
• Part of the sphere appears to be missing because of the rectangular grid system used by the computer.
COMPUTER GRAPHICS
• The much better picture here was produced by a computer using the parametric equations found in Example 4.
PARAMETRIC REPRESENTATIONS

Example 5

• Find a parametric representation for the cylinderx2 + y2 = 4 0 ≤ z≤ 1
• The cylinder has a simple representation r = 2 in cylindrical coordinates.
• So, we choose as parameters θand zin cylindrical coordinates.
PARAMETRIC REPRESENTATIONS

Example 5

• Then the parametric equations of the cylinder are x = 2 cos θy = 2 sin θz = z
• where:
• 0 ≤ θ≤ 2π
• 0 ≤z≤ 1
PARAMETRIC REPRESENTATIONS

Example 6

• Find a vector function that represents the elliptic paraboloid z = x2 + 2y2
• If we regard x and y as parameters, then the parametric equations are simply x = xy = yz = x2 + 2y2and the vector equation isr(x, y) = xi + yj + (x2 + 2y2) k
PARAMETRIC REPRESENTATIONS
• In general, a surface given as the graph of a function of x and y—an equation of the form z = f(x, y)—can always be regarded as a parametric surface by:
• Taking x and y as parameters.
• Writing the parametric equations as x = xy = yz = f(x, y)
PARAMETRIZATIONS
• Parametric representations (also called parametrizations) of surfaces are not unique.
• The next example shows two ways to parametrize a cone.
PARAMETRIZATIONS

Example 7

• Find a parametric representation for the surface that is, the top half of the cone z2 = 4x2 + 4y2
PARAMETRIZATIONS

E. g. 7—Solution 1

• One possible representation is obtained by choosing x and y as parameters:x = xy = y
• So, the vector equation is:
PARAMETRIZATIONS

E. g. 7—Solution 2

• Another representation results from choosing as parameters the polar coordinates r and θ.
• A point (x, y, z) on the cone satisfies: x = r cos θy = r sin θ
PARAMETRIZATIONS

E. g. 7—Solution 2

• So, a vector equation for the cone is r(r, θ) = r cos θi + r sin θj + 2rkwhere:
• r≥ 0
• 0 ≤ θ≤ 2π
PARAMETRIZATIONS
• For some purposes, the parametric representations in Solutions 1 and 2 are equally good.
• In certain situations, though, Solution 2 might be preferable.
PARAMETRIZATIONS
• For instance, if we are interested only in the part of the cone that lies below the plane z = 1, all we have to do in Solution 2 is change the parameter domain to: 0 ≤ r ≤ ½ 0 ≤θ≤ 2π
SURFACES OF REVOLUTION
• Surfaces of revolution can be represented parametrically and thus graphed using a computer.
SURFACES OF REVOLUTION
• For instance, let’s consider the surface S obtained by rotating the curve y = f(x) a≤ x ≤ babout the x-axis, where f(x) ≥ 0.
SURFACES OF REVOLUTION
• Let θbe the angle of rotation as shown.
SURFACES OF REVOLUTION

Equations 3

• If (x, y, z) is a point on S, then x = xy = f(x) cos θz = f(x) sin θ
SURFACES OF REVOLUTION
• Thus, we take x and θas parameters and regard Equations 3 as parametric equations of S.
• The parameter domain is given by: a≤ x ≤ b 0 ≤ θ≤ 2π
SURFACES OF REVOLUTION

Example 8

• Find parametric equations for the surface generated by rotating the curve y = sin x, 0 ≤ x ≤ 2π, about the x-axis.
• Use these equations to graph the surface of revolution.
SURFACES OF REVOLUTION

Example 8

• From Equations 3,
• The parametric equations are: x = xy = sin x cos θz = sin x sin θ
• The parameter domain is: 0 ≤ x ≤ 2π 0 ≤θ≤ 2π
SURFACES OF REVOLUTION

Example 8

• Using a computer to plot these equations and rotate the image, we obtain this graph.
SURFACES OF REVOLUTION
• We can adapt Equations 3 to represent a surface obtained through revolution about the y- or z-axis.
• See Exercise 30.
TANGENT PLANES
• We now find the tangent plane to a parametric surface S traced out by a vector function r(u, v) = x(u, v) i + y(u, v) j + z(u, v) kat a point P0 with position vector r(u0, v0).
TANGENT PLANES
• Keeping u constant by putting u = u0, r(u0, v) becomes a vector function of the single parameter v and defines a grid curve C1lying on S.
TANGENT PLANES

Equation 4

• The tangent vector to C1 at P0 is obtained by taking the partial derivative of r with respect to v:
TANGENT PLANES
• Similarly, keeping v constant by putting v = v0, we get a grid curve C2 given by r(u, v0) that lies on S.
TANGENT PLANES

Equation 5

• Its tangent vector at P0 is:
SMOOTH SURFACE
• If ru x rv is not 0, then the surface is called smooth(it has no “corners”).
• For a smooth surface, the tangent plane is the plane that contains the tangent vectors ru and rv , and the vector ru x rv is a normal vector to the tangent plane.
TANGENT PLANES

Example 9

• Find the tangent plane to the surface with parametric equations x = u2y = v2z = u + 2vat the point (1, 1, 3).
TANGENT PLANES

Example 9

• We first compute the tangent vectors:
TANGENT PLANES

Example 9

• Thus, a normal vector to the tangent plane is:
TANGENT PLANES

Example 9

• Notice that the point (1, 1, 3) corresponds to the parameter values u = 1 and v = 1.
• So, the normal vector there is: –2 i + 4 j + 4 k
TANGENT PLANES

Example 9

• Therefore, an equation of the tangent plane at (1, 1, 3) is: –2(x – 1) – 4(y – 1) + 4(z – 3) = 0orx + 2y – 2z + 3 = 0
TANGENT PLANES
• The figure shows the self-intersecting surface in Example 9 and its tangent plane at (1, 1, 3).
SURFACE AREA
• Now, we define the surface area of a general parametric surface given by Equation 1.
SURFACE AREAS
• For simplicity, we start by considering a surface whose parameter domain Dis a rectangle, and we divide it into subrectangles Rij.
SURFACE AREAS
• Let’s choose (ui*, vj*) to be the lower left corner of Rij.
PATCH
• The part Sij of the surface S that corresponds to Rij is called a patchand has the point Pij with position vector r(ui*, vj*) as one of its corners.
SURFACE AREAS
• Let ru* = ru(ui*, vj*) and rv* = rv(ui*, vj*) be the tangent vectors at Pij as given by Equations 5 and 4.
SURFACE AREAS
• The figure shows how the two edges of the patch that meet at Pij can be approximated by vectors.
SURFACE AREAS
• These vectors, in turn, can be approximated by the vectors Δuru* and Δvrv* because partial derivatives can be approximated by difference quotients.
• So, we approximate Sij by the parallelogram determined by the vectors Δuru* and Δvrv*.
SURFACE AREAS
• This parallelogram is shown here.
• It lies in the tangent plane to S at Pij.
SURFACE AREAS
• The area of this parallelogram is: So, an approximation to the area of S is:
SURFACE AREAS
• Our intuition tells us that this approximation gets better as we increase the number of subrectangles.
• Also, we recognize the double sum as a Riemann sum for the double integral
• This motivates the following definition.
SURFACE AREAS

Definition 6

• Suppose a smooth parametric surface S is:
• Given byr(u, v) = x(u, v) i + y(u, v) j + z(u, v) k (u, v) D
• Covered just once as (u, v) ranges throughout the parameter domain D.
SURFACE AREAS

Definition 6

• Then, the surface areaof S iswhere:
SURFACE AREAS

Example 10

• Find the surface area of a sphere of radius a.
• In Example 4, we found x = a sin Φ cos θ, y = a sin Φ sin θ,z = a cos Φwhere the parameter domain is:D = {(Φ, θ) | 0 ≤ Φ ≤ π, 0 ≤ θ ≤ 2π)
SURFACE AREAS

Example 10

• We first compute the cross product of the tangent vectors:
SURFACE AREAS

Example 10

SURFACE AREAS

Example 10

• Thus,
• since sin Φ ≥ 0 for 0 ≤ Φ≤π.
SURFACE AREAS

Example 10

• Hence, by Definition 6, the area of the sphere is:
SURFACE AREA OF THE GRAPH OF A FUNCTION
• Now, consider the special case of a surface S with equation z = f(x, y), where (x, y) lies in Dand f has continuous partial derivatives.
• Here, we take x and y as parameters.
• The parametric equations are:x = xy = yz = f(x, y)
GRAPH OF A FUNCTION

Equation 7

• Thus,and
GRAPH OF A FUNCTION

Equation 8

• Thus, we have:
GRAPH OF A FUNCTION

Formula 9

• Then, the surface area formula in Definition 6 becomes:
GRAPH OF A FUNCTION

Example 11

• Find the area of the part of the paraboloid z = x2 + y2 that lies under the plane z = 9.
• The plane intersects the paraboloid in the circle x2 + y2 = 9, z = 9
GRAPH OF A FUNCTION

Example 11

• Therefore, the given surface lies above the disk Dwith center the origin and radius 3.
GRAPH OF A FUNCTION

Example 11

• Using Formula 9, we have:
GRAPH OF A FUNCTION

Example 11

• Converting to polar coordinates, we obtain:
SURFACE AREA
• The question remains:
• Is our definition of surface area (Definition 6) consistent with the surface area formula from single-variable calculus (Formula 4 in Section 8.2)?
SURFACE AREA
• We consider the surface S obtained by rotating the curve y = f(x), a≤ x ≤ b about the x-axis, where:
• f(x) ≥ 0.
• f’ is continuous.
SURFACE AREA
• From Equations 3, we know that parametric equations of S are:
• x = xy = f(x) cos θz = f(x) sin θa≤ x ≤ b 0 ≤ θ≤ 2π
SURFACE AREA
• To compute the surface area of S, we need the tangent vectors
SURFACE AREA
• Hence,because f(x) ≥ 0.
SURFACE AREA
• Thus, the area of S is:
SURFACE AREA
• This is precisely the formula that was used to define the area of a surface of revolution in single-variable calculus (Formula 4 in Section 8.2).