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Chapter 14: Q. 12 (page 1106)

Translate $d r$ into terms of $r (t)$ .

Short Answer

Expert verified

The $d r$ in term of $r (t)$ is $dr = ||r' (t)|| d t$ .

Step by step solution

01

Step 1. Given Information

Translate $d s$ 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 of $f (x, y, z)$ along C is

$\int_{C} f (x, y, z) dr = \int_{a}^{b} f (x (t), y (t), z (t)) ||r' (t)|| d t$

After comparing

data-custom-editor="chemistry" $d r = ||r' (t)|| d t$

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

The current through a certain passage of the San Juan Islands in Washington State is given by

$F = <F_{1} (x, y), F_{2} (x, y)> = <0, 1.152 - 0.8 x^{2}> .$

Consider a disk R of radius 1 mile and centered on this region. Denote the boundary of the disk by localid="1650455155947" $\partial R$ .

(a) Compute localid="1650455166518" $\iint_{R} (\frac{\partial F_{2}}{\partial x} - \frac{\partial F_{1}}{\partial y}) d A$ .
(b) Show that $\int_{\partial R} F \cdot n d s = \int_{0}^{2 π} (1.152 - 0.8 \cos^{2} θ) \sin θ d θ .$ Conclude that Green's Theorem is valid for the current in this area of the San Juan Islands.
(c) What do the integrals from Green's Theorem tell us about this region of the San Juan Islands?

$Use the vector field F (x, y) = x^{2} e^{y} i + \cos (x) \sin (y) j and Green ’ s Theorem to write the line integral of F (x, y) about the unit circle, traversed counterclockwise, as a double integral . Do not evaluate the integral .$

Use the same vector field as in Exercise 13, and compute the k-component of the curl of F(x, y).

$Compute the curl of the vector fields in Exercises 23 - 28 . F (x, y, z) = {xe}^{yz} i + {ye}^{xz} j + {ze}^{xy} k$

Calculus of vector-valued functions: Calculate each of the following.

$\frac{d}{d t} (r (t)), where r (t) = 3 \cos^{2} t i + 5 t j + \frac{t}{t^{2} + 1} k$

See all solutions

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