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A helix can be described parametrically as Find the Tangent, Normal and Binormal vectors to the helix. Explain why these three vectors, called the Frenet-Serret frame, form an orthonormal basis for R3. Show that the curvature is kappa = a/a2+c2 and the torsion is tau = c/a2+c2 Explain in your own words what curvature and torsion are in a physical sense. This should be in some detail, perhaps with examples or pictures. Find the Tangent vector for the generalised exponential helix, Consider the curve z = x4 - 2x2. Write the curve parametrically in 3 dimensions. Using the linear transformation for rotation, rotate this curve around the z axis. Convert the parametric equation for the surface to an equation of the form z = f(x, y). Hint: make use of the identity cos2 theta + sin2 theta = 1. Find the critical points of the resulting surface. Determine the nature of the points using the second derivative test . If the test fails, explain why (this may take some research into the more technical definition of local extrema). Describe and carry out your own alternative method to determine the nature of the critical point/s for which the test fails. Consider the definite integral Give a convincing argument that The above double integral can not be evaluated using standard integration techniques. Convert the double integral to polar coordinates. Evaluate the integral to show that I = .

Show transcribed image text A helix can be described parametrically as Find the Tangent, Normal and Binormal vectors to the helix. Explain why these three vectors, called the Frenet-Serret frame, form an orthonormal basis for R3. Show that the curvature is kappa = a/a2+c2 and the torsion is tau = c/a2+c2 Explain in your own words what curvature and torsion are in a physical sense. This should be in some detail, perhaps with examples or pictures. Find the Tangent vector for the generalised exponential helix, Consider the curve z = x4 - 2x2. Write the curve parametrically in 3 dimensions. Using the linear transformation for rotation, rotate this curve around the z axis. Convert the parametric equation for the surface to an equation of the form z = f(x, y). Hint: make use of the identity cos2 theta + sin2 theta = 1. Find the critical points of the resulting surface. Determine the nature of the points using the second derivative test . If the test fails, explain why (this may take some research into the more technical definition of local extrema). Describe and carry out your own alternative method to determine the nature of the critical point/s for which the test fails. Consider the definite integral Give a convincing argument that The above double integral can not be evaluated using standard integration techniques. Convert the double integral to polar coordinates. Evaluate the integral to show that I = .