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.
25% of 400 is 100.
It means that there are 100 red jelly beans and 300 belly jeans of a different colour in the bag.
Your chance of picking a jelly bean out of the bag which isn't red is 300/400. 300/400 is equal to 75% therefore the chance of not selecting a red jelly bean out of the bag is 0.75.
Answer:
Step-by-step explanation:
cos θ=12/16=3/4=0.75
θ=cos^{-1}(0.75)≈41.4°
Answer:
three (3)
Step-by-step explanation:
4x is the first term
-4 is the second term
3y is the third term
(X+2)(X+10) heres the factored form