- Analytic geometry in space. Vector algebra in R3. Scalar, cross and triple product
and their geometric meaning. Lines, planes and quadric surfaces in space including
the standard equations for cones, ellipsoids, paraboloids and hyperboloids.
- Functions of several variables.Graphs and level curves and surfaces. Limits and
continuity. Properties of the continuous functions on a closed bounded domain.
Partial derivatives. The plane tangent to graph of the function. Differentiability, the
total differential and the linear approximation. Differentiability implies continuity.
The chain rule. The gradient vector and the directional derivative. Tangent plane and
the normal line to a surface at a point.
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- Maxima and minima for functions of several variables. Higher-order partial
derivatives and differentials. Taylor’s formula. Local extrema and saddle points.
Necessary conditions for local maxima and minima of a differentiable function.
Sufficient conditions for local maxima and minima via the Hessian. Global extrema in
closed bounded sets. Lagrange Multipliers.
- Double integrals . Double integrals on rectangles. Connection with the volume.
Properties and evaluation of double integrals in non-rectangular domains. Iterated
integrals and change of order of integration. Change of variables formula for the
double integral and the Jacobian. Double integrals in polar coordinates. Applications
of the change of variables formula to the computation of area.