The temperature (in degrees Celsius) at a point x , y on a metal plate in the x y -plane is T x , y = x y 1 + x 2 + y 2 (a) Find the rate of change of temperature at 1 , 1 in the direction of a = 2 i − j . (b) An ant at 1 , 1 wants to walk in the direction in which the temperature drops most rapidly. Find a unit vector in that direction.
The temperature (in degrees Celsius) at a point x , y on a metal plate in the x y -plane is T x , y = x y 1 + x 2 + y 2 (a) Find the rate of change of temperature at 1 , 1 in the direction of a = 2 i − j . (b) An ant at 1 , 1 wants to walk in the direction in which the temperature drops most rapidly. Find a unit vector in that direction.
The temperature (in degrees Celsius) at a point
x
,
y
on a metal plate in the
x
y
-plane
is
T
x
,
y
=
x
y
1
+
x
2
+
y
2
(a) Find the rate of change of temperature at
1
,
1
in the direction of
a
=
2
i
−
j
.
(b) An ant at
1
,
1
wants to walk in the direction in which the temperature drops most rapidly. Find a unit vector in that direction.
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.
Subtracting the two equations, find a vector equation for the curve of intersection between y= 4x2+(3/4)z2 and y-1= 3x2+(1/2)z2 for x>0. Find and simplify the tangential component of acceleration for your curve.
The concentration of salt in a fluid at
is given by
mg/cm
. You are at the point
.
(a) In which direction should you move if you want the concentration to increase the fastest?
Direction:
(Give your answer as a vector.)
(b) You start to move in the direction you found in part (a) at a speed of
cm/sec. How fast is the concentration changing?
Rate of change =
HINT: The rate of change of the perceived concentration F(x,y,z), by the Chain Rule, equals the dot product of the gradient vector of F and the velocity of the "particle". To find it, we need to know the norms (magnitudes) of both vectors and the angle between them. In this problem the angle is known.
(14) The straight line L passes through the origin O and is in the
direction i+mj. The straight line L' passes through the point A
whose position vector is ai and is in the direction i+m'j. Write
down the vector equations of L and L' and find the position
vector of their point of intersection.
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