Observe that the given vector field is a gradient field:
Let
, so that



Integrating the first equation with respect to
, we get

Differentiating this with respect to
gives

Now differentiating
with respect to
gives

Putting everything together, we find a scalar potential function whose gradient is
,

It follows that the curl of
is 0 (i.e. the zero vector).
Answer:
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Explanation:
The gain in gravitational potential energy of the mass = 100 kg m^2 / s^2.
<u>Explanation:</u>
Gravitational potential energy is an energy in which an object possesses because of its position in a gravitational field. The most common use of gravitational potential energy is for an object near the Earth's surface where the gravitational acceleration can be assumed to be constant at about 9.8 m/s2.
The formula for the gravitational potential energy is the product of mass, gravity, and height.
GPE = m *g *h
where m represents mass in kg,
g represents the gravity,
h represents the height.
GPE = 5 * 10 * 2 = 100 kg m^2 / s^2.
For a star that has the same apparent brightness as Alpha Centauri A ( 2.7×10−8watt/m2 is mathematically given as
L=2.7*10^30w
<h3> What is its luminosity?</h3>
Generally, the equation for the luminosity is mathematically given as
L=4*\pi^2*b
Therefore
L=4*\pi^2*b
L=4* \pi *(2.83*10^{18})*2.7*10^{-8}
L=2.7*10^30
In conclusion, the luminosity
L=2.7*10^30w
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