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.
Answer:
1 3/8 ft.
Step-by-step explanation:
<em>The cabinet's center will be at the same place as the wall's center</em>. The wall's center is at its <em>3 1/8 ft mark</em>. The cabinet's center will also be at the wall's 3 1/8 mark. This means that both sides of the cabinet is <em>1 3/4 around the center line</em> (if my wording is vague, look at the image). So, we have to find <em>3 1/8 - 1 3/4</em>, which is 1 3/8 ft.
Answer:
Reason SAS postulate
Step-by-step explanation:
The two triangles must be congruent.
Reason: SAS postulate
The answer is 52.5 million.
Sorry for being late.
Answer:
A) f(x) = |x|
B) f(x) = -|x+3| + 3
Step-by-step explanation:
Absolute value functions
The general form of an absolute value function is f(x) = a|x-b| +c
a indicates the 'amplitude' or multiplier
b indicates horizontal shift
c indicates vertical shift
The turning point of f(x) = |x| is (0, 0)
a = y/x when b = 0 and c = 0
A) f(x) = |x|
Since the turning point is still (0, 0) there is no horizontal nor vertical shift, meaning b = 0 and c = 0
We have the point (2, 2), thus a = 2/2 = 1
f(x) = a|x-b| + c
f(x) = |x|
B) f(x) = -|x+3| + 3
The turning point is at (-3, 3) b = -3 and c = +3
We have the point (3, -3), thus a = y/x = -3/3 = -1
f(x) = a|x-b| + c
f(x) = (-1) (|x-(-3)|) + (+3)
f(x) = -|x+3| + 3