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Chapter 14: Q. 11 (page 1119)

Given a smooth surface S described as a function z = f(x, y), calculate the upwards-pointing normal vector for S.

Short Answer

Expert verified

Therefore, the required upwards-pointing normal vector for a smooth surface S described as a function z = f(x, y) will be,

n=-fxi-fyj+k

Step by step solution

01

Step 1. Given 

The function given is z = f(x, y),

02

Step 2. Find rx and ry.

By above step , surface S is parametrized by as follows.

r(x,y)=⟨x,y,f(x,y)⟩.Thenrx=∂∂xr(x,y)=∂∂x⟨x,y,f(x,y)⟩=1,0,∂f∂x=1,0,fxandry=∂∂yr(x,y)=∂∂y⟨x,y,f(x,y)⟩=0,1,∂f∂y=0,1,fy

03

Step 3.  Find n=rx×ry

n=rx×ry=1,0,fx×0,1,fy=ijk10fx01fy=0⋅fy−fx⋅1i−1⋅fy−fx⋅0j+[1⋅1−0⋅0]k=−fxi−fyj+k.

Here, the z-component of this normal vector is 1 which is positive, so this normal vector is upwards-pointing normal vector.

Therefore, the required upwards-pointing normal vector for a smooth surface S described as a function z = f(x, y) will be,

n=-fxi-fyj+k

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Most popular questions from this chapter

Give a formula for a normal vector to the surface S determined by y = g(x,z), where g(x,z) is a function with continuous partial derivatives.

Q. True/False: Determine whether each of the statements that follow is true or false. If a statement is true, explain why. If a statement is false, provide a counterexample.

(a) True or False: Stokes’ Theorem asserts that the flux of a vector field through a smooth surface with a smooth boundary is equal to the line integral of this field about the boundary of the surface.

(b) True or False: Stokes’ Theorem can be interpreted as a generalization of Green’s Theorem.

(c) True or False: Stokes’ Theorem applies only to conservative vector fields.

(d) True or False: Stokes’ Theorem is always used as a way to evaluate difficult surface integrals.

(e) True or False: Stokes’ Theorem can be interpreted as a generalization of the Fundamental Theorem of Line Integrals.

(f) True or False: If F(x, y ,z) is a conservative vector field, then Stokes’ Theorem and Theorem 14.12 together give an alternative proof of the Fundamental Theorem of Line Integrals for simple closed curves.

(g) True or False: Stokes’ Theorem can be interpreted as a generalization of the Fundamental Theorem of Calculus.

(h) True or False: Stokes’ Theorem can be used to evaluate surface area .

What does it mean to say that a vector field is conservative?

Find the work done by the vector field

F(x,y)=cosx-3yexi+sinxsinyj

in moving an object around the periphery of the rectangle with vertices (0,0),(2,0),(2,π), and (0,π), starting and ending at (2,π).

How would you show that a given vector field in ℝ3 is not conservative?
See all solutions

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