If the electric potential at a point x , y in the x y -plane is V x , y , then the electric intensity vector at the point x , y is E = − ∇ V x , y . Suppose that V x , y = e − 2 x cos 2 y . (a) Find the electric intensity vector at π / 4 , 0 . (b) Show that at each point in the plane, the electric potential decreases most rapidly in the direction of the vector E .
If the electric potential at a point x , y in the x y -plane is V x , y , then the electric intensity vector at the point x , y is E = − ∇ V x , y . Suppose that V x , y = e − 2 x cos 2 y . (a) Find the electric intensity vector at π / 4 , 0 . (b) Show that at each point in the plane, the electric potential decreases most rapidly in the direction of the vector E .
If the electric potential at a point
x
,
y
in the
x
y
-plane
is
V
x
,
y
,
then the electric intensity vector at the point
x
,
y
is
E
=
−
∇
V
x
,
y
.
Suppose that
V
x
,
y
=
e
−
2
x
cos
2
y
.
(a) Find the electric intensity vector at
π
/
4
,
0
.
(b) Show that at each point in the plane, the electric potential decreases most rapidly in the direction of the vector
E
.
Quantities that have magnitude and direction but not position. Some examples of vectors are velocity, displacement, acceleration, and force. They are sometimes called Euclidean or spatial vectors.
A charged particle begins at rest at the origin. Suddenly, a force causes the particleto accelerate according to the vector function a(t) = ⟨ sin(t) , 6t , 2cos(t)⟩Find functions for the velocity, speed and position of the particle at time t
The force on a particle is described by 10x°
+ 6
at a point x along the x-axis. Find the work done
in moving the particle from the origin to x = 5.
The motion of a point on the circumference of a rolling wheel of radius 2 feet is described by the
vector function
r(t) = 2(23t sin (23t))i + 2(1 - cos(23t))j
-
Find the velocity vector of the point.
v(t)
=
Find the acceleration vector of the point.
a(t)
=
Find the speed of the point.
s(t)
=
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