The integral is path-independent if we can find a scalar function <em>f</em> such that grad(<em>f</em> ) = <em>A</em>. This requires
Take the first PDE and integrate both sides with respect to <em>x</em> to get
where <em>g</em> is assumed to be a function of <em>y</em> alone. Differentiating this with respect to <em>x</em> gives
which would mean <em>g</em> is *not* a function of only <em>y</em>, but also <em>x</em>, contradicting our assumption. So the integral is path-dependent.
Parameterize the unit circle (call it <em>C</em>) by the vector function,
with <em>t</em> between 0 and 2π.
Note that this parameterization takes <em>C</em> to have counter-clockwise orientation; if we compute the line integral of <em>A</em> over <em>C</em>, we can multiply the result by -1 to get the value of the integral in the opposite, clockwise direction.
Then
and the (counter-clockwise) integral over <em>C</em> is
and so the integral in the direction we want is -2π.
By the way, that the integral doesn't have a value of 0 is more evidence of the fact that the integral is path-dependent.
The sum of the series is 999/1000 .
<h3>What is a Series ?</h3>
A series is a sequence of expression in a certain pattern.
The series of n terms is given
nth term of the series is given by
On simplification it can be written as
The sum of terms from 1 to 9 is given by
∑ ( )
=
= (1/1³) - (1/10³)
= 999/1000
Therefore the sum of the series given is 999/1000
To know more about Series
brainly.com/question/15415793
#SPJ1
Answer:
The inequality m ≥ 2 has infinity many solutions because 2 and any number to the right of 2 on the number line can be substituted for m to make this inequality true.
Answer:
mmm idgl
Step-by-step explanation:
Answer:
$145.80
Step-by-step explanation:
The original cost of the product was $90. There was a 162% markup. In order to get the selling price, I like to take away 100 percent and simply add original price+(original price*62%). After taking away 100%, I am left with 62%. I then multiply the original cost ($90) by the remaining percent (62%), getting 55.8. Following the steps that I originally wrote, I now add 90 to 55.8, getting 145.8.