[Math] Line integral of vector field to find work done when particle moves along a line.

integration

A particle moves along the path $(t,t^2,t^3)$, $0 \leq t \leq 1$, and experiences a constant unit force parallel to vector $(-1,0,1)$. What is the work done by the force?

Attempt:

I am studying for the GRE Mathematics Subject Test, and I don't know any physics. While preparing for the test, I learned how to integrate line integrals over a vector field.

i.e. $ F(x,y) = M dx + N dy$ is a vector field. $r(t) = (x(t),y(t))$ is a parameterized curve where $a \leq t \leq b$. Then, the line integral over vector field F is

$$\int_a^b \! F(x(t),y(t)) \cdot (x'(t),y'(t)) \, \mathrm{d} t$$

So, I know that the integral to solve for my question is:

$$\int_0^1 \! F(t,t^2,t^3) \cdot (1,2t,3t^3) \, \mathrm{d} t$$

Question:

How do I find the equation for the vector field, $F(x,y,z)$? Or is my approach wrong?

Best Answer

You are looking for a vector field $F \colon \mathbb{R}^3 \to \mathbb{R}^3$ such that

  1. $|F|=1$ (constant unit force)
  2. $F$ is parallel to $(-1,0,1)$.

Then $F(x,y,z)=\frac{1}{\sqrt{2}}(-1,0,1)$.

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