Tangent Plane to a Surface

In document Vector Calculus (Page 83-86)

2.3 Tangent Plane to a Surface

In the previous section we mentioned that the partial derivatives fx and fy can be thought of as the rate of change of a functionz=f(x,y) in the positivexandydirections, respectively. Recall that the derivative d ydx of a function y=f(x) has a geometric meaning, namely as the slope of the tangent line to the graph of f at the point (x,f(x)) in R2. There is a similar geometric meaning to the partial derivatives fx and fy of a function z= f(x,y): given a point (a,b) in the domainDof f(x,y), the trace of the surface described byz=f(x,y) in the plane y=bis a curve inR3 through the point (a,b,f(a,b)), and the slope of the tangent line Lxto that curve at that point is fx(a,b). Similarly, fy(a,b) is the slope of the tangent line

Lyto the trace of the surfacez=f(x,y) in the planex=a(see Figure 2.3.1).

y z x 0 (a,b) D Lx b (a,b,f(a,b)) slope=∂f∂x(a,b) z=f(x,y)

(a)Tangent lineLxin the planey=b

y z x 0 (a,b) D Ly a (a,b,f(a,b)) slope=∂f∂y(a,b) z=f(x,y)

(b)Tangent lineLyin the planex=a Figure 2.3.1 Partial derivatives as slopes

Since the derivative d ydx of a function y=f(x) is used to find the tangent line to the graph off (which is a curve inR2), you might expect that partial derivatives can be used to define a tangent planeto the graph of a surface z=f(x,y). This indeed turns out to be the case. First, we need a definition of a tangent plane. The intuitive idea is that a tangent plane “just touches” a surface at a point. The formal definition mimics the intuitive notion of a tangent line to a curve.

Definition 2.4. Letz=f(x,y) be the equation of a surfaceS inR3, and letP=(a,b,c) be a point onS. Let T be a plane which contains the pointP, and letQ=(x,y,z) represent a generic point on the surface S. If the (acute) angle between the vector −−→PQ and the plane T approaches zero as the point Q approaches P along the surface S, then we call T the

tangent planetoSatP.

Note that since two lines inR3determine a plane, then the two tangent lines to the surface z=f(x,y) in the xand ydirections described in Figure 2.3.1 are contained in the tangent plane at that point, if the tangent plane exists at that point. The existence of those two

tangent lines does not by itself guarantee the existence of the tangent plane. It is possible that if we take the trace of the surface in the planexy=0 (which makes a 45◦angle with the positive x-axis), the resulting curve in that plane may have a tangent line which is not in the plane determined by the other two tangent lines, or it may not have a tangent line at all at that point. Luckily, it turns out4that if xf and fy exist in a region around a point (a,b) and are continuous at (a,b) then the tangent plane to the surface z=f(x,y) will exist at the point (a,b,f(a,b)). In this text, those conditions will always hold.

y z x 0 (a,b,f(a,b)) z=f(x,y) T Lx Ly

Figure 2.3.2 Tangent plane Suppose that we want an equation of the tangent planeT

to the surfacez=f(x,y) at a point (a,b,f(a,b)). LetLxand Ly be the tangent lines to the traces of the surface in the planes y=b andx=a, respectively (as in Figure 2.3.2), and suppose that the conditions forT to exist do hold. Then the equation forTis

A(xa)+B(yb)+C(zf(a,b))=0 (2.4) wheren=(A,B,C) is a normal vector to the planeT. Since

T contains the linesLxandLy, then all we need are vectorsvxandvythat are parallel toLx andLy, respectively, and then letn=vx×××vy.

x z 0 vx=(1, 0, ∂f ∂x(a,b)) ∂f ∂x(a,b) 1 Figure 2.3.3 Since the slope of Lx is

∂f

∂x(a,b), then the vector vx=(1, 0,

∂f

∂x(a,b)) is

parallel toLx (sincevx lies in thexz-plane and lies in a line with slope ∂f

∂x(a,b) 1 =

∂f

∂x(a,b). See Figure 2.3.3). Similarly, the vector

vy=(0, 1,

∂f

∂y(a,b)) is parallel toLy. Hence, the vector

n=vx×××vy= ¯ ¯ ¯ ¯ ¯ ¯ ¯ i j k 1 0 fx(a,b) 0 1 fy(a,b) ¯ ¯ ¯ ¯ ¯ ¯ ¯ = −xf(a,b)i∂f ∂y(a,b)j+k

is normal to the planeT. Thus the equation ofT is

xf(a,b) (xa)fy(a,b) (yb)+zf(a,b)=0. (2.5)

Multiplying both sides by1, we have the following result:

The equation of the tangent plane to the surfacez=f(x,y) at the point (a,b,f(a,b)) is

∂f

∂x(a,b) (xa)+

∂f

∂y(a,b) (yb)z+f(a,b)=0 (2.6)

2.3 Tangent Plane to a Surface 77

Example 2.13.Find the equation of the tangent plane to the surface z=x2+y2at the point (1, 2, 5).

Solution: For the function f(x,y)=x2+y2, we have ∂fx=2xand fy=2y, so the equation of the tangent plane at the point (1, 2, 5) is

2(1)(x1)+2(2)(y2)z+5=0 , or 2x+4yz5=0.

In a similar fashion, it can be shown that if a surface is defined implicitly by an equation of the formF(x,y,z)=0, then the tangent plane to the surface at a point (a,b,c) is given by the equation ∂F ∂x(a,b,c) (xa)+ ∂F ∂y(a,b,c) (yb)+ ∂F ∂z(a,b,c) (zc)=0. (2.7)

Note that formula (2.6) is the special case of formula (2.7) whereF(x,y,z)=f(x,y)z.

Example 2.14.Find the equation of the tangent plane to the surfacex2+y2+z2=9 at the point (2, 2,1).

Solution:For the functionF(x,y,z)=x2+y2+z29, we have Fx =2x, Fy =2y, and Fz =2z, so the equation of the tangent plane at (2, 2,1) is

2(2)(x2)+2(2)(y2)+2(1)(z+1)=0 , or 2x+2yz9=0.

Exercises

A

For Exercises 1-6, find the equation of the tangent plane to the surface z= f(x,y) at the pointP.

1. f(x,y)=x2+y3,P=(1, 1, 2) 2. f(x,y)=x y,P=(1,1,1)

3. f(x,y)=x2y,P=(1, 1, 1) 4. f(x,y)=xey,P=(1, 0, 1)

5. f(x,y)=x+2y,P=(2, 1, 4) 6. f(x,y)=px2+y2,P=(3, 4, 5)

For Exercises 7-10, find the equation of the tangent plane to the given surface at the point P. 7. x42+y92+16z2=1,P=³1, 2,2 p 11 3 ´ 8. x2+y2+z2=9,P=(0, 0, 3) 9. x2+y2z2=0,P=(3, 4, 5) 10. x2+y2=4,P=(p3, 1, 0)

In document Vector Calculus (Page 83-86)