Surface area and surface integrals. (Sect. 16.5) Review ...

Surface area and surface integrals. (Sect. 16.5)

Review: Arc length and line integrals. Review: Double integral of a scalar function. Explicit, implicit, parametric equations of surfaces. The area of a surface in space.

The surface is given in parametric form. The surface is given in explicit form.

Review: Arc length and line integrals

The integral of a function f : [a, b] R is

b

n

a

f (x) dx = lim

n

f (xi) x.

i =0

The arc length of a curve r : [t0, t1] R3 in space is

t1

st1,t0 = |r (t)| dt.

t0

The integral of a function f : R3 R along a curve

t1

r : [t0, t1] R3 is f ds = f r(t) |r (t)| dt.

C

t0

The circulation of a function F : R3 R3 along a curve

t1

r : [t0, t1] R3 is F ? u ds = F r(t) ? r (t) dt.

C

t0

The flux of a function F : {z = 0} R3 {z = 0} R3 along

a loop r : [t0, t1] {z = 0} R3 is F = F ? n ds.

C

Surface area and surface integrals. (Sect. 16.5)

Review: Arc length and line integrals. Review: Double integral of a scalar function. Explicit, implicit, parametric equations of surfaces. The area of a surface in space.

The surface is given in parametric form. The surface is given in explicit form.

Review: Double integral of a scalar function

The double integral of a function f : R R2 R on a region R R2, which is the volume under the graph of f and above

the z = 0 plane, and is given by

nn

R

f

dA

=

lim

n

i =0

j =0

f

(xi, yj) x

y .

The area of a flat surface R R2 is the particular case f = 1, that is, A(R) = dA.

R

We will show how to compute: The area of a non-flat surface in space. (Today.) The integral of a scalar function f on a surface is space. The flux of a vector-valued function F on a surface in space.

Surface area and surface integrals. (Sect. 16.5)

Review: Arc length and line integrals. Review: Double integral of a scalar function. Explicit, implicit, parametric equations of surfaces. The area of a surface in space.

The surface is given in parametric form. The surface is given in explicit form.

Explicit, implicit, parametric equations of surfaces

Review: Curves on R2 can be defined in:

Explicit form, y = f (x); Implicit form, F (x, y ) = 0; Parametric form, r(t) = x(t), y (t) . The vector r (t) = x (t), y (t) is tangent to the curve.

Review: Surfaces in R3 can be defined in:

Explicit form, z = f (x, y ); Implicit form, F (x, y , z) = 0; Parametric form, r(u, v ) = x(u, v ), y (u, v ), z(u, v ) . Two vectors tangent to the surface are

ur(u, v ) = ux(u, v ), uy (u, v ), uz(u, v ) , v r(u, v ) = v x(u, v ), v y (u, v ), v z(u, v ) .

Explicit, implicit, parametric equations of surfaces

Example

Find a parametric expression for the cone z = tangent vectors.

x2 + y 2, and two

Solution: Use cylindrical coordinates: x = r cos(), y = r sin(), z = z. Parameters of the surface: u = r , v = . Then

x(r , ) = r cos(), y (r , ) = r sin(), z(r , ) = r . Using vector notation, a parametric equation of the cone is

r(r , ) = r cos(), r sin(), r .

Two tangent vectors to the cone are r r and r, r r = cos(), sin(), 1 , r = -r sin(), r cos(), 0 .

Explicit, implicit, parametric equations of surfaces

Example

Find a parametric expression for the sphere x2 + y 2 + z2 = R2, and two tangent vectors. Solution: Use spherical coordinates: x = cos() sin(), y = sin() sin(), z = cos(). Parameters of the surface: u = , v = .

x = R cos() sin(), y = R sin() sin(), z = R cos(). Using vector notation, a parametric equation of the cone is

r(, ) = R cos() sin(), sin() sin(), cos() .

Two tangent vectors to the paraboloid are r and r, r = R - sin() sin(), cos() sin(), 0 ,

r = R cos() cos(), sin() cos(), - sin() .

Surface area and surface integrals. (Sect. 16.5)

Review: Arc length and line integrals. Review: Double integral of a scalar function. Explicit, implicit, parametric equations of surfaces. The area of a surface in space.

The surface is given in parametric form. The surface is given in explicit form.

The area of a surface in parametric form

Theorem

Given a smooth surface S with parametric equation

r(u, v ) = x(u, v ), y (u, v ), z(u, v ) for u [u0, u1] and v [v0, v1]

is given by

u1 v1

A(S) =

|ur ? v r| dv du.

u0 v0

z S = { r ( u, v ) }

u= 0 v= 0

du r y

x

dv r

Remark: The function

d = |ur ? v r| dv du.

represents the area of a small region on the surface. This is the generalization to surfaces of the arc-length formula for the length of a curve.

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