Geometry and Vectors
- Use the parallelogram law to add geometric vectors. Resolve geometric
vectors into components parallel to coordinate axes.
- Perform the operations of vector addition and scalar multiplication,
and interpret them geometrically.
- Use the dot product to calculate magnitude of a vector, angle
between vectors, and projection of one vector on another.
- Find and use direction angles and direction cosines of a vector.
- Use parametric equations for plane curves and space curves.
- Use and convert between parametric and symmetric equations for a
straight line.
- Find a tangent line at a point on a parametric curve; compute the
length of a parametric curve; compute the area of the surface
generated by revolving a plane curve about an axis.
- Compute velocity, unit tangent and acceleration vectors along a
parametric curve; resolve acceleration into tangential and normal
components and compute curvature.
- Use and interpret geometrically the standard and symmetric equations
for a plane.
- Use the cross product; interpret the cross product geometrically and
as the area of a parallelogram; interpret the vector triple product
as the volume of a parallelopiped.
- Recognize cylinders and quadric surfaces from their Cartesian
equations.
- Use cylindrical and spherical coordinates, and convert among these
two and rectangular coordinates.
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The Derivative in n-Space
- Represent a function of two variables as the graph of a surface;
sketch level curves.
- Calculate partial derivatives and the gradient.
- Use the gradient to find tangent planes, directional derivatives and
linear approximations. Interpret the gradient geometrically.
- Use the Chain Rule.
- Find and classify critical points of functions, using the second
derivative test.
- Use Lagrange's method to maximize or minimize a function subject to
constraints.
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The Integral in n-Space
- State the definition of the integral of a function over a rectangle.
- Use iterated integrals to evaluate integrals over planar regions, and
to calculate volume.
- Set up and evaluate double integrals in polar coordinates.
- Set up and evaluate integrals to compute surface area.
- Set up and evaluate triple integrals in Cartesian coordinates.
- Use double and triple integrals to compute moments, center of mass,
and moments of inertia.
- Use cylindrical and spherical coordinates; change coordinates from
rectangular to cylindrical or spherical or the reverse.
- Set up and evaluate triple integrals in cylindrical and spherical
coordinates.
- Change the order of variables in multiple integrals.
- Carry out change of variables in multiple integrals.
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Vector Calculus
- Calculate the curl and divergence of a vector field.
- Set up and evaluate line integrals of scalar functions or vector
fields along curves.
- Recognize conservative vector fields, and apply the fundamental
theorem for line integrals of conservative vector fields.
- State and apply Green's Theorem.
- Set up and evaluate integrals to compute the area of parametric
surfaces.
- Set up and evaluate surface integrals, surface area, and the flux of
a vector field through a surface.
- State and apply the Divergence Theorem.
- State and apply Stokes' Theorem.
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