Calculus Answers

Determine the square of the arc element for the curvilinear coordinate system (u,v,w) whose coordinates are related to the Cartesian coordinates as follows: x = 3u+v-w, y=u +2v+2w, z = 2u - v-w.

Obtain the curl of the following vector field:: A ^vector = (rho ^3 ×e^cap subscript p + rho ×z×e^cap subscript phi + rho ×z ×sin phi e^cap subscript z)

Show that for any scalar field phi: nabla^vector ×(phi nnabla^vector phi) =00

Evaluate (nabla/del ^vector) (r^3)(r^vector).

Obtain the directional derivative for a scalar field phi(x, y, z) = 3(x^2)y - (y^3)(z^2) at the point (1, -2, -1) in the direction i^cap+j^cap+k^cap.

Use the mean value theorem to show that x < sin-¹x , for x > 0

Sketch the graph of the function f defined by 4 3 f(x) = x + 8x , clearly giving all the
properties used in it.

Obtain a unit tangent vector to any point on the curve defined by the parameteric equations: x = sin 3t, y = 2cos 3t, z = 4t.

Find the center, vertices, and foci of the ellipse with equation 2x2 + 7y2 = 14.

Which of the following statements are true or false? Give reasons for your answers. (2 × 5 =10)
i) The function R R →:f , given by | x1x|n)x(f 2 ++= l is neither even nor odd. ii) ∫ − = 0 x 22 xsindt)tsin( dx d .
iii) The area enclosed by the x-axis and the curve x cosy = over the interval ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ π π − 2 3 , 2
is 0. iv) If f and g are functions over R such that g f + is continuous, then f must be continuous. v) 0 2yx = +− is a tangent to the curve 2 3 ) 2yx()yx( +−=+ at ) 1,1( − .