Question

Find the curvature of ~r (t) = t 2 ,sin(t) − t cos(t), cos(t) tsin(t) at any t > 0.

Answer 1

To find the curvature of ~r(t) = t^2, sin(t) - t cos(t), cos(t) tsin(t), find ~r'(t) and ~r''(t) and apply the formula κ = ||~r'(t) x ~r''(t)|| / ||~r'(t)||^3.

Step-by-step explanation:

The curvature, denoted by κ, of a curve ~r(t) can be found using the formula κ = ||~r'(t) x ~r''(t)|| / ||~r'(t)||^3. To find the curvature of ~r(t) = t^2, sin(t) - t cos(t), cos(t) tsin(t), we need to find ~r'(t) and ~r''(t) and then apply the formula.

Step 1: Find ~r'(t) by differentiating each component of ~r(t) with respect to t.

Step 2: Find ~r''(t) by differentiating each component of ~r'(t) with respect to t.

Step 3: Calculate κ using the formula κ = ||~r'(t) x ~r''(t)|| / ||~r'(t)||^3.