Partial Differentiation of Function of Several Variables. Differentiation of Implicit Functions, Theorem and Applications. Jacobians. Differentiation of a Vector Function of Several Variables. Total Differential, Tangent Plane to Surface, the Tangent Vector. Curvilinear Coordinates; Plane Polar, Cylindrical and Spherical Co-ordinates. Multiple Integrals. Line Integrals, Multiple Surface and Volume Integral. Gradient, Divergence and Curl. The Theorems of Green, Gauss, and Stokes

The Chain Rule for Derivatives The Chain Rule for Partial Derivatives

State Fundamental theorem of Gradient, Divergence and Stokes.

Let f (x, y, z) = sin(xy + z), and P be the point (0, −2, π3 ).(a) (6 pts) Compute ∇f (x, y, z).(b) (2 pts) At P , find the direction along which f obtains maximum directional derivative.(c) (4 pts) Calculate the directional derivative ∂f∂u (P ), where u is a unit vector making an angleπ6 with the gradient ∇f (P ).(d) (3 pts) The level surface f (x, y, z) = √32 defines z implicitly as a function of x and y near P .Compute ∂z∂x at P .

Suppose that f is a function of two variables (y and z) only.!Show that the gradient ∇ f = (∂ f /∂ y)ˆy + (∂ f /∂z)ˆz transforms as a vector un-der rotations, Eq. 1.29. [Hint: (∂ f /∂ y) = (∂ f /∂ y)(∂ y/∂ y) + (∂ f /∂z)(∂z/∂ y),and the analogous formula for ∂ f /∂z. We know that y = y cos φ + z sin φ andz = −y sin φ + z cos φ; “solve” these equations for y and z (as functions of yand z), and compute the needed derivatives ∂ y/∂ y, ∂z/∂ y, etc.]

Implicit Differentiation