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Advanced Engineering Mathematics, International Student Edition

Peter V. O'Neil

Chapter 12

Vector Differential Calculus - all with Video Answers

Educators

Section 1

Vector Functions of One Variable

Problem 1

In each of Problems 1 through 8, compute the requested derivative (a) by carrying out the vector operation and differentiating the resulting vector or scalar, and (b) by using one of the differentiation rules (1) through (5) stated at the end of this section.
$\mathbf{F}(t)=\mathbf{i}+3 t^2 \mathbf{j}+2 t \mathbf{k}, f(t)=4 \cos (3 t) ;(d / d t)[f(t) \mathbf{F}(t)]$

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Problem 2

In each of Problems 1 through 8, compute the requested derivative (a) by carrying out the vector operation and differentiating the resulting vector or scalar, and (b) by using one of the differentiation rules (1) through (5) stated at the end of this section.
$\mathbf{F}(t)=t \mathbf{i}-3 t^2 \mathbf{k}, \mathbf{G}(t)=\mathbf{i}+\cos (t) \mathbf{k}$; $(d / d t)[\mathbf{F}(t) \cdot \mathbf{G}(t)]$

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Problem 3

In each of Problems 1 through 8, compute the requested derivative (a) by carrying out the vector operation and differentiating the resulting vector or scalar, and (b) by using one of the differentiation rules (1) through (5) stated at the end of this section.
$\mathbf{F}(t)=t \mathbf{i}+\mathbf{j}+4 \mathbf{k}, \mathbf{G}(t)=\mathbf{i}-\cos (t) \mathbf{j}+t \mathbf{k}$; $(d / d t)[\mathbf{F}(t) \times \mathbf{G}(t)]$

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Problem 4

In each of Problems 1 through 8, compute the requested derivative (a) by carrying out the vector operation and differentiating the resulting vector or scalar, and (b) by using one of the differentiation rules (1) through (5) stated at the end of this section.
$\mathbf{F}(t)=\sinh (t) \mathbf{j}-t \mathbf{k}, \mathbf{G}(t)=t \mathbf{i}+t^2 \mathbf{j}-t^2 \mathbf{k}$; $(d / d t)[\mathbf{F}(t) \times \mathbf{G}(t)]$

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Problem 5

In each of Problems 1 through 8, compute the requested derivative (a) by carrying out the vector operation and differentiating the resulting vector or scalar, and (b) by using one of the differentiation rules (1) through (5) stated at the end of this section.
$\mathbf{F}(t)=t \mathbf{i}-\cosh (t) \mathbf{j}+e^t \mathbf{k}, f(t)=1-2 t^3$; $(d / d t)[f(t) \mathbf{F}(t)]$

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Problem 6

In each of Problems 1 through 8, compute the requested derivative (a) by carrying out the vector operation and differentiating the resulting vector or scalar, and (b) by using one of the differentiation rules (1) through (5) stated at the end of this section.
$\mathbf{F}(t)=t \mathbf{i}-t \mathbf{j}+t^2 \mathbf{k}, \mathbf{G}(t)=\sin (t) \mathbf{i}-4 \mathbf{j}+t^3 \mathbf{k}$; $(d / d t)[\mathbf{F}(t) \cdot \mathbf{G}(t)]$

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Problem 7

In each of Problems 1 through 8, compute the requested derivative (a) by carrying out the vector operation and differentiating the resulting vector or scalar, and (b) by using one of the differentiation rules (1) through (5) stated at the end of this section.
$\mathbf{F}(t)=-9 \mathbf{i}+t^2 \mathbf{j}+t^2 \mathbf{k}, \mathbf{G}(t)=e^t \mathbf{i} ;(d / d t)[\mathbf{F}(t) \times \mathbf{G}(t)]$

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Problem 8

In each of Problems 1 through 8, compute the requested derivative (a) by carrying out the vector operation and differentiating the resulting vector or scalar, and (b) by using one of the differentiation rules (1) through (5) stated at the end of this section.
$\mathbf{F}(t)=-4 \cos (t) \mathbf{k}, \mathbf{G}(t)=-t^2 \mathbf{i}+4 \sin (t) \mathbf{k}$; $(d / d t)[\mathbf{F}(t) \cdot \mathbf{G}(t)]$

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06:06

Problem 9

In each of Problems 9, 10, and 11, (a) write the position vector and tangent vector for the curve whose parametric equations are given, (b) find a length function $s(t)$ for the curve, (c) write the position vector as a function of $s$, and (d) verify that the resulting position vector has a derivative of length 1 .
$x=\sin (t), y=\cos (t), z=45 t ;(0 \leq t \leq 2 \pi)$

Nick Johnson
Nick Johnson
Numerade Educator
01:27

Problem 10

In each of Problems 9, 10, and 11, (a) write the position vector and tangent vector for the curve whose parametric equations are given, (b) find a length function $s(t)$ for the curve, (c) write the position vector as a function of $s$, and (d) verify that the resulting position vector has a derivative of length 1 .
$x=y=z=t^3 ;(-1 \leq t \leq 1)$

Sajin Shajee
Sajin Shajee
Numerade Educator
01:27

Problem 11

In each of Problems 9, 10, and 11, (a) write the position vector and tangent vector for the curve whose parametric equations are given, (b) find a length function $s(t)$ for the curve, (c) write the position vector as a function of $s$, and (d) verify that the resulting position vector has a derivative of length 1 .
$x=2 t^2, y=3 t^2, z=4 t^2 ;(1 \leq t \leq 3)$

Sajin Shajee
Sajin Shajee
Numerade Educator
00:48

Problem 12

Let $\mathbf{F}(t)=x(t) \mathbf{i}+y(t) \mathbf{j}+z(t) \mathbf{k}$. Suppose $x, y$ and $z$ are differentiable functions of $t$. Think of $\mathbf{F}(t)$ as the position function of a particle moving along a curve in 3-space. Suppose $\mathbf{F} \times \mathbf{F}^{\prime}=\mathbf{O}$. Prove that the particle always moves in the same direction.

Hoan Nguyen
Hoan Nguyen
Numerade Educator