Concept explainers
(a)
To find:
Solution:
Explanation:
1) Concept:
If
2) Given:
3) Calculation:
If
Integrate
Notice that the constant of integration is a constant with respect to
Now differentiate above equation with respect to
Comparing (2), and (4)
Integrating with respect to
Hence (4) becomes
Differentiate with respect to
Comparing this equation with (1)
Integrate with respect to
Therefore, (6) becomes,
Conclusion:
(b)
To evaluate:
Solution:
Explanation:
1) Concept:
Fundamental theorem of line integral:
Let
2) Given:
3) Calculation:
C is a smooth curve with initial point
So, by using concept,
Since
Therefore,
Therefore,
From the answer of part (a),
Therefore,
Conclusion:
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- Consider the R − R 2 function r defined by r (t) = t, t2 ; t ∈ [−3, 3] . (a) Determine the vector derivative r 0 (1) (b) Sketch the curve r together with the vector r 0 (1), in order to illustrate the geometric meaning of the vector derivative.arrow_forwardFind the domain of the vector-valued function. (Enter your answer using interval notation.) r(t) = F(t) + G(t), where F(t) = cos ti – sin tj + 8Vik, G(t) = 2 cos ti + sin tjarrow_forwardFind r′(t), r(t0), and r′(t0) for the given value of t0 = 3. Then sketch the curve represented by the vector-valued function r(t) = (1 − t2)i + tj, and sketch the vectors r(t0) and r′(t0).arrow_forward
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