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Chapter 14: Q. 11 (page 1106)

Translate ds into terms of r(t).

Short Answer

Expert verified

The ds in term of r(t)isds=r'(t)dt

Step by step solution

01

Step 1. Given Information

Translate ds into terms of data-custom-editor="chemistry" r(t).

02

Step 2. Let C be a curve in ℝ3 with a smooth parametrization r(t)=(x(t), y(t), z(t)) for t∈[a,b].

Then the integral of f(x,y,z) along C is

鈭�Cf(x,y,z)ds=鈭�abf(x(t),y(t),z(t))r'(t)dt

After comparing

ds=r'(t)dt

The integral along C is sometimes referred to as the integral with respect to arc length.

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Most popular questions from this chapter

What is the difference between the graphs of

G(x,y)=xi+yjandF(x,y)=鈭�xi+(-y)j

F(x,y,z)=zcosyzj+zsinyzk, where S is the portion of the plane with equation 2x鈭�8y鈭�10z=42 that lies on the positive side of the rectangle with corners0,-蟺,0(0,蟺,0),(0,蟺,蟺),(0,鈭�蟺,蟺)in theyz-plane.

Integrate the given function over the accompanying surface in Exercises 27鈥�34.
f(x,y,z)=x2-y+3z, where Sis the portion of the plane with equation 2x-6y+3z=1whose preimage in the xz plane is the region bounded by the coordinate axes and the lines with equations z = 4 and x = z.

Compute n for the surface S in Exercise 12.

Find the work done by the vector field

F(x,y)=x3y2i+(y-x)j

in moving an object around the triangle with vertices (1,1),(2,2), and (3,1), starting and ending at (2,2).
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