#### 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

We talk about basic geometry in higher dimensions.

#### Drawing a sphere

Learn how to draw a sphere.

#### Vectors

Vectors are lists of numbers that denote direction and magnitude.

#### The dot product

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

#### The cross product

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

#### Lines and curves in space

Vector-valued functions are parameterized curves.

#### Calculus and vector-valued functions

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

#### 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.

#### 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

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

#### Level sets

We introduce level sets.

#### Continuity

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

#### 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

We find tangent planes.

#### Differentiability

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

#### 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 vector

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

#### Taylor polynomials

We introduce Taylor polynomials for functions of several variables.

We will get to know some basic quadric surfaces.

#### Drawing paraboloids

Learn how to draw an elliptic and a hyperbolic paraboloid.

#### Maxima and minima

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

#### Constrained optimization

We learn to optimize surfaces along and within given paths.

#### Lagrange multipliers

We give a new method of finding extrema.

#### Integrals over trivial regions

We study integrals over basic regions.

#### Integrals with trivial integrands

We study integrals over general regions by integrating $1$.

#### 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.

#### 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 fields

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

#### Line integrals

We accumulate vectors along a path.

#### 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.

#### 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.

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