Understanding Multivariable Calculus: Problems, Solutions, and Tips

Reviewing key concepts from basic calculus; a brief overview of what is to be learned; applying distance and midpoint formulas to three-dimensional objects; observing some of the new curiosities unique to functions of more than one variable.

What makes a function "multivariable"; beginning with definitions; how these new functions behave as one applies familiar concepts of minimum and maximum values; using graphics and other tools to observe their interactions with the xy-plane.

Applying fundamental definitions of calculus to multivariable functions, starting with their limits; how these limits become complicated as one approaches them; using this to derive the definition of a versatile new tool, the partial derivative.

Discovering the new dimensions of second partial derivatives; differentiating either twice with respect to x or y, or once each with respect to x and y; considering Laplace's equation to see what makes a function "harmonic".

Combining the differential and chain rule from elementary calculus; how to generalize them to functions of more than one variable; how the total differential can be used to approximate ?z over small intervals without calculating the exact values.

The ability to find extreme values for optimization is one of the most powerful consequences of differentiation; defining the Extreme Value Theorem for multivariable functions; using it to identify relative extrema using a "second partials test".

Exploring multivariable optimization by using the Extreme Value Theorem on closed and bounded regions; finding absolute minimum and maximum values across bounded regions of a function; applying these concepts to a real-world problem.

Applying techniques of optimization to curve-fitting; exploring a statistical tool yielded by multivariable calculus; beginning with the Least Squares Regression Line that yields the best fit to a set of points; applying it to a real-life problem.

Beginning the study of vectors in three-dimensional space; extrapolating vector notation and formulas for magnitude from the familiar equations for two dimensions; learning to derive the parametric equations of a line parallel to a direction vector.

Taking the cross product of two vectors, yielding a third vector perpendicular to both; the properties of this new vector; how to define the triple scalar product; using it to evaluate the volume of a parallelepiped.

Defining a plane using the tools acquired so far; learning about projections of one vector onto another; finding the angle between two planes; using vector projections to find the distance between a point and a plane.

Applying what one has learned to curved surfaces; beginning with the equation of a sphere; finding the recognizable parabolas and other 2-D shapes that lie hidden in new vector equations; observing surfaces of revolution in three-dimensional space.

Defining vector-valued functions and their derivatives; various formulas relating to arc length; applying these definitions to position, velocity and acceleration vectors; differentiating them using a surprisingly simple method.

Examining Johannes Kepler's laws of planetary motion; applying vector-valued functions to Newton's second law of motion and his law of gravitation; how Newton is able to take laws Kepler derives from observation and prove them using calculus.

Building on the knowledge of multivariable differentiation with gradient vectors; using them to determine directional derivatives; discovering a unique property of the gradient vector and its relationships with level curves and surfaces.

Utilizing the gradient to find normal vectors to a surface; how these vectors interplay with standard functions to determine the tangent plane to a surface at a given point; starting with tangent planes to level surfaces; analyzing the result.

The method of Lagrange multipliers; using this theorem and some simple algebra to optimize functions subject not just to boundaries, but to constraints given by multivariable functions; applying this tool to a real-world cost-optimization example.

Observing the simplicity of Lagrange multipliers firsthand by re-examining an optimization problem from an earlier episode; exploring one of the many uses of constrained optimization in the world of physics by deriving Snell's Law of Refraction.

Exploring integration, the other side of calculus in three dimensions; starting off with iterated integrals, an intuitive and simple approach that merely adds an extra step and a slight twist to one-dimensional integration.

Learning to integrate multivariable functions; the double integral has many of the same properties as its one-dimensional counterpart; evaluating these over a region R bounded by variable constraints; extrapolating a single-variable formula.

Transforming Cartesian functions f(x, y) into polar coordinates defined by r and θ; getting familiar with surfaces defined by this new coordinate system; how these coordinates can be used to derive simple and elegant solutions from certain integrals.

Applying new methods of evaluating integrals over a region to the realm of physics; learning the formulas for mass and moments of mass for a planar lamina of variable density; finding the center of mass for these regions.

Bringing another fundamental idea of calculus into three dimensions by expanding arc lengths into surface areas; reviewing arc length and surfaces of revolution; the formulas for surface area and the differential of surface area over a region.

Taking the next step to triple integrals, which can be used to find the volume of a solid in space; extrapolating the density of planar lamina to volumes defined by triple integrals; evaluating density in the form of mass per unit of volume.

Exploring the immediate extension of double integrals into volumes with cylindrical coordinates, moving from a surface defined by (r,?) to a cylindrical volume with an extra parameter defined by (r,?,z); using these conversions to simplify problems.

Observing how spherical coordinates often yield more useful information in a more concise format than other coordinate systems; how they are essential in evaluating triple integrals over a spherical surface.

An introduction to vector fields; their role in describing gravitational and electric fields; learning the definition of a conservative vector field using the gradient vector; calculating the potential of a conservative vector field on a plane.

Using the gradient vector to find the curl and divergence of a field, both of which describe the rotation and movement of a particle in these fields; exploring the line integral, which is used to evaluate a density function over a curved path.

One of the most important applications of the line integral is its ability to calculate work done on an object as it moves along a path in a force field, learning how vector fields make the orientation of a path significant.

Generalizing the fundamental theorem of calculus; the key properties of curves in space; learning what makes a curve smooth, piecewise-smooth, simple and closed; manipulating curves to reveal new, simpler methods of evaluating some line integrals.

Using one of the most important theorems in multivariable calculus to observe how a line integral can be equivalent to an often more-workable area integral; why the line integral around a closed curve is equal to zero in a conservative vector field.

Transforming difficult line integrals quickly and efficiently into more approachable double integrals; learning an alternative form of Green's theorem that generalizes to some important upcoming theorems.

Extending one's understanding of surfaces by defining them in terms of parametric equations; learning to graph parametric surfaces; learning to calculate surface area.

Discovering the surface integral and a special case known as the flux integral; evaluating the surface integral as a double integral; continuing the study of fluid mechanics by utilizing this integral to evaluate flux in a vector field.

The Divergence theorem, a hallmark of multivariable calculus, combines flux and triple integrals; discovering the divergence of a fluid; calling upon the gradient vector to define how a surface integral over a boundary can give the volume of a solid.

Developing Stokes's theorem, the third capstone relationship between the new integrals of multivariable calculus; seeing how a line integral equates to a surface integral; connections to Maxwell's equations for electric and magnetic fields.