Ximera tutorial

How to use Ximera

This course is built in Ximera.

How is my work scored?

We explain how your work is scored.

Working in two and three dimensions

Working in two and three dimensions

We talk about basic geometry in higher dimensions.

Drawing a sphere

Learn how to draw a sphere.

Vector-valued functions



Vectors are lists of numbers that denote direction and magnitude.

Dot products

The dot product

The dot product measures how aligned two vectors are with each other.

Cross products

The cross product

The cross product is a special way to multiply two vectors in three-dimensional space.

Lines and curves in space

Lines and curves in space

Vector-valued functions are parameterized curves.

Calculus and vector-valued functions

Calculus and vector-valued functions

With one input, and vector outputs, we work component-wise.

Motion and paths in space

Motion and paths in space

We interpret vector-valued functions as paths of objects in space.

Parameterizing by arc length

We find a new description of curves that trivializes arc length computations.

Normal vectors

Unit tangent and unit normal vectors

We introduce two important unit vectors.

Planes in space

We discuss how to find implicit and explicit formulas for planes.

Parametric plots

Tangent and normal vectors can help us make interesting parametric plots.

Drawing a torus

Learn how to draw a torus.

Functions of several variables

Functions of several variables

Functions of several variables

We introduce functions that take vectors or points as inputs and output a number.

Level sets

We introduce level sets.

Continuity of functions of several variables


We investigate what continuity means for real-valued functions of several variables.

Partial derivatives and the gradient vector

Partial derivatives

We introduce partial derivatives and the gradient vector.

Approximating with the gradient

We use the gradient to approximate values for functions of several variables.

Tangent planes and differentiability

Tangent planes

We find tangent planes.


We introduce differentiability for functions of several variables and find tangent planes.

The directional derivative and the chain rule

The directional derivative

We introduce a way of analyzing the rate of change in a given direction.

The chain rule

We investigate the chain rule for functions of several variables.

Interpreting the gradient

Interpreting the gradient vector

The gradient is the fundamental notion of a derivative for a function of several variables.

Taylor polynomials

Taylor polynomials

We introduce Taylor polynomials for functions of several variables.

Quadric surfaces

Quadric surfaces

We will get to know some basic quadric surfaces.

Drawing paraboloids

Learn how to draw an elliptic and a hyperbolic paraboloid.

Maximums and minimums

Maxima and minima

We see how to find extrema of functions of several variables.

Constrained optimization

Constrained optimization

We learn to optimize surfaces along and within given paths.

Lagrange multipliers

Lagrange multipliers

We give a new method of finding extrema.

Multiple integrals

Integrals over trivial regions

We study integrals over basic regions.

Integrals with trivial integrands

We study integrals over general regions by integrating .

Common coordinates

Polar coordinates

We integrate over regions in polar coordinates.

Cylindrical coordinates

We integrate over regions in cylindrical coordinates.

Spherical coordinates

We integrate over regions in spherical coordinates.

Computations and interpertations

Surface area

We compute surface area with double integrals.

Mass, moments, and center of mass

We use integrals to model mass.

Computations and interpretations

We practice more computations and think about what integrals mean.

Vector-valued functions of several variables

Vector fields

Vector fields

We introduce the idea of a vector at every point in space.

Line integrals

Line integrals

We accumulate vectors along a path.

Green’s Theorem

Curl and Green’s Theorem

Green’s Theorem is a fundamental theorem of calculus.

Green’s Theorem as a planimeter

A planimeter computes the area of a region by tracing the boundary.

Divergence and Green’s Theorem

Divergence measures the rate field vectors are expanding at a point.

The shape of things to come

Surface integrals

We generalize the idea of line integrals to higher dimensions.

Drawing a Mobius strip

Learn how to draw a Möbius strip.

Divergence theorem

We introduce the divergence theorem.

Stokes’ theorem

We introduce Stokes’ theorem.

Grad, Curl, Div

We explore the relationship between the gradient, the curl, and the divergence of a vector field.

You can download a Certificate as a record of your successes.