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Chapter 14: Q. 12 (page 1106)

Translate dr into terms of r(t).

Short Answer

Expert verified

The dr in term of r(t) is dr=r'(t)dt.

Step by step solution

01

Step 1. Given Information

Translate ds into terms of 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 off(x,y,z)along C is

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

After comparing

data-custom-editor="chemistry" dr=r'(t)dt

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

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Most popular questions from this chapter

Compute dS for your parametrization in Exercise 7.

Let a, b, and c be nonzero constants. Find a general formula for the area of the portion of the plane with equation ax+by+cz=kthat lies above a rectangle [伪,尾]脳[纬,未] in thexy-plane.

Examples: Construct examples of the thing(s) described in the following. Try to find examples that are different than any in the reading.

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Give two different unit normal vectors to S at the point r(u0,v0).

(c) A smooth surface that can be smoothly parametrized as r(x,z)=x,f(x),z.

Give an example of a vector field whose orientation does not affect the outcome of Stokes鈥� Theorem.

Do the vectors in the range of F(x,y)=xi+yjpoint towards or away from the origin?
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