Suppose that over a certain region of space the electrical potential v is given by the following equation. v(x, y, z) = 3x2 − 3xy + xyz (a) find the rate of change of the potential at p(3, 6, 5) in the direction of the vector v = i + j − k. (b) in which direction does v change most rapidly at p? (c) what is the maximum rate of change at p?
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Ответ:
(a) 18
(b) in the direction of the gradient vector
(c)![6\sqrt{35}](/tex.php?f=6\sqrt{35})
explanation:
the function
is a differentiable function of x,y,z since it is a polynomial. therefore, the directional derivative at the point
in the direction of a unit vector
is the dot product
(the derivative is the rate of change, so this is the rate of change in the direction of the unit vector
at the point
.)
we need a unit vector in the direction of vector
. the magnitude of
is
so a unit vector in the direction of
is
now we have to calculate the partial derivatives of v at (3,6,5). as we have
, we calculate partial derivative with respect to x of v by treating y and z as constants and differentiating with respect to x:
similarly, we calculate![v_y(3,6,5)](/tex.php?f=v_y(3,6,5))
and we calculate![v_z(3,6,5)](/tex.php?f=v_z(3,6,5))
(a)
we have
(b)
we want a unit vector
that maximizes the value of
. so here, we fix the point (x,y,z) and find
. since we have the gradient vector defined as
we can rewrite the formula for the directional derivative as
this is a dot product of two vectors. so if
is the angle between these two vectors (where
), then we have
but clearly |u| = 1 since it is a unit vector, so we have
the maximum of this occurs when
has its maximum, and for
, the maximum of
occurs when
. this means the unit vector
has to be in the same direction as the gradient vector
.
and since
, the maximum value of the directional derivative at a point is
.
we calculated before that
so v increases fastest in the direction that gradient vector.
(c)as discussed before, maximum value of the directional derivative at a point is
. so the maximum rate of change of v at point (3,6,5) is
or since 1260 = 36*35, we have
as our final answer.
Ответ:
The formula for acceleration is:
a = (Vf - Vi) / t
Where:
a = acceleration
Vf = final velocity
Vi = initial velocity
t = time
(Note that the formula can also be written using V, Velocity, and V0, Velocity Nought/zero. Its the same as final and initial velocity, with initial being nought)
We have all we need to do the problem now:
Vf = 20 m/s
Vi = 5 m/s
t = 3s
a = (20 - 5) / 3
a = 5 m/s^2