Calculate Fly's Path: gradients & curves in Space

[tandoorichicken]

"The temperature of space is given by $\phi (x,y,z) = xy + xz$. A fly is flying in space and at each point (x,y,z) of its journey it flies in the direction $\mathbf{F} (x,y,z)$ in which the rate of increase of temperature is maximum.

(a.) Calculate F(x,y,z).
(b.) Find the curve along which the fly travels if it starts at the point (1,0,1).
"

For part (a), the direction in which the temperature changes is the same as that of the gradient of $\phi$, so
$$\mathbf{F}(x,y,z) = \nabla\phi (x,y,z) = (y+z)\mathbf{\hat{i}}+x\mathbf{\hat{j}}+x\mathbf{\hat{k}}$$

It is part (b) that is giving me trouble. I just don't know where to begin. I'm sure if I had that much I could figure out the rest on my own.

Any help appreciated! (Final in four days)

 

[0rthodontist]

In part (b) you want to find the curve c(t) such that

F(c(t)) = c'(t)

and c(0) = (1, 0, 1)

 

[HallsofIvy]

"The temperature of space is given by $\phi (x,y,z) = xy + xz$. A fly is flying in space and at each point (x,y,z) of its journey it flies in the direction $\mathbf{F} (x,y,z)$ in which the rate of increase of temperature is maximum.

(a.) Calculate F(x,y,z).
(b.) Find the curve along which the fly travels if it starts at the point (1,0,1).
"

For part (a), the direction in which the temperature changes is the same as that of the gradient of $\phi$, so
$$\mathbf{F}(x,y,z) = \nabla\phi (x,y,z) = (y+z)\mathbf{\hat{i}}+x\mathbf{\hat{j}}+x\mathbf{\hat{k}}$$

It is part (b) that is giving me trouble. I just don't know where to begin. I'm sure if I had that much I could figure out the rest on my own.

Any help appreciated! (Final in four days)

$$\mathbf{F}(x,y,z) = \nabla\phi (x,y,z) = (y+z)\mathbf{\hat{i}}+x\mathbf{\hat{j}}+x\mathbf{\ hat{k}}$$
points in the direction of flight, which is the same direction as the velocity vector. Since speed is not relevant, you can take "t" to be any parameter you like and then
$$\frac{dx}{dt}= \mathbf{F}_x= y+ z$$
$$\frac{dy}{dt}= \mathbf{F}_y= x$$
$$\frac{dz}{dt}= \mathbf{F}_z= x$$
You need to solve those three equations with initial condition x(0)= 1, y(0)= 0, z(0)= 1. That will give you parametric equations (with t as the parameter) for the path of flight.