# Preclass Notes

MATH 126

## Table of contents

- Week 1 Monday (12.1)
- Week 1 Wednesday (12.2)
- Week 1 Friday (12.3)
- Week 2 Monday (12.4, 12.5)
- Week 2 Wednesday (12.5)
- Week 2 Friday (12.6)
- Week 3 Monday (13.1)
- Week 3 Wednesday (13.2)
- Week 3 Friday (13.3a)
- Week 4 Monday (13.3b)
- Week 4 Friday (13.4)
- Week 5 Monday (14.1)
- Week 5 Wednesday (14.7)
- Week 6 Wednesday (15.1)
- Week 6 Friday (15.2)
- Week 7 Monday (10.3)
- Week 7 Wednesday (15.3)
- Week 7 Friday (15.4)
- Week 8 Wednesday (Taylor Notes 1)
- Week 8 Friday (Taylor Notes 2)
- Week 9 Monday (Taylor Notes 3)
- Week 9 Wednesday (Taylor Notes 4)
- Week 9 Friday (Taylor Series 5)
- Week 10 Wednesday

## Week 1 Monday (12.1)

- Three-dimensional spaces
- We need an origin - a reference point. Begin with two axes that are perpendicular. The third axis must extend perpendicular to both of the existing axes.
- Right hand rule: helps you determine in which direction a third axis points.
We can describe every point space with some set of coordinates \((a, b, c)\)/\((x, y, z)\).

- \(\mathbb{R} = (-\infty, \infty)\) - the set of real unmbers.
- Set notation. All points in 3D form:

Set theory: {\text{bag of possibilities} \text{ rule(s)}}$$ Surfaces: $${(x, y, z) x, y, z \in \mathbb{R}, z = 3}\(gives you the\)x-y\(plane hovering at level\)z = 3$$. - Three important planes: at \(x = 0, y = 0, z = 0\).
- Just stating the rule \(z = 3\) (the equations) is also acceptable.

## Week 1 Wednesday (12.2)

- Vector: depicted as an arrow, with a length as its magnitude, an initial point, and a terminal point
- Adding two vectors. The sum of two vectors is the vector from the initial point of one vector to the terminal point of the other vector.
- Scalar multiplication. Scalar - a real number. Preserves parallel properties.
- A position vector is a vector extending from the origin to a specific point.
- Components - number of dimensions of a vector.
- Length/magnitude of a vector can be calculated using Pythagorean theorem
- Standard basis vectors - unit vectors along the dimensions.
- You can decompose any vector into a linear sum of the standard basis vectors.

## Week 1 Friday (12.3)

- Dot product/scalar product/inner product: element-wise weighted sum. For a vector \(\vec{a} = \langle a_1, a_2, ..., a_n \rangle\) and \(\vec{b} = \langle b_1, b_2, ..., b_n \rangle\), the dot product is \(\vec{a} \cdot \vec{b} = a_1 b_1 + a_2 b_2 + ... + a_n b_n\).
- The input is a set of vectors, but the output is a scalar. The dot product produces a different mathematical structure.
- \[\vec{v} \cdot \vec{v} = \| \vec{v} \|^2\]
- Theorem: if the angle between two vectors \(\vec{a}\) and \(\vec{b}\) is \(\theta\), then \(\vec{a} \cdot \vec{b} = \| \vec{a} \| \| \vec{b} \| \cos \theta\).
- Alternatively, \(\cos \theta = \frac{\vec{a} \cdot \vec{b}}{\| \vec{a} \| \| \vec{b} \| }\)

- Two vectors are orthogonal if and only if \(\vec{v} \cdot \vec{w} = 0\).
- A zero-vector is orthogonal to every other vector.

- Projection - what does some \(\vec{b}\) do to \(\vec{a}\)?
- Scalar projection of \(\vec{b}\) onto \(\vec{a}\): \(\text{comp}_{\vec{a}} \vec{b} = \frac{ \vec{a} \cdot \vec{b}}{\| \vec{a} \|}\). This
*component*is a real number. Exploits a trigonometric relationship. - To obtain the vector projection, scale the vector to a unit vector and multiply by the scalar projection.

- Scalar projection of \(\vec{b}\) onto \(\vec{a}\): \(\text{comp}_{\vec{a}} \vec{b} = \frac{ \vec{a} \cdot \vec{b}}{\| \vec{a} \|}\). This

## Week 2 Monday (12.4, 12.5)

Section 12.4

- The cross-product holds only for three-dimensional vectors.
- Determinant method.
- The resulting vector \(\vec{v} = \vec{a} \times \vec{b}\) is orthogonal to both \(\vec{a}\) and \(\vec{b}\).
- If you swap the order, you will get the negative product.
- \(\| \vec{a} \times \vec{b} \| = \| \vec{a} \| \| \vec{b} \| \sin \theta\), where \(\theta\) is the smallest angle between \(\vec{a}\) and \(\vec{b}\). This is a good test for parallelity (equals 0).
- Area of parallelogram: \(\| \vec{a} \times \vec{b} \|\)

Section 12.5 Part I

- Vector equations: equations in which we want to equate a vector on the LHS to a vector on the RHS.
- Map a vector to a point
- You can multiply a vector by a scalar and it will remain on the same parallel line.
- Multiply the direction vector - make it shorter, longer, switch the direction, etc.
- If \(t \in \mathbb{R}\), we get all points on that line.
- \[\vec{r} = \vec{r_0} + t\vec{v}\]
- \(\vec{r_0}\) - position vector
- \(t \in \mathbb{R}\) - parameter
- \(\vec{v}\) - direction vector

- Sometimes, we write vectors vertically instead of horizontally.
- Properties:
- Two lines are parallel if their direction vectors are parallel
- Two lines intersect if they have a point \((x, y, z)\) in common
- Two lines are skew if they are not parallel and do not intersect.

## Week 2 Wednesday (12.5)

- Vector form of a line: \(\vec{r} = \vec{r}_0 + t\vec{v}\)
- \(\vec{v}\) is the direction vector
- \(\vec{r}_0\) is the position vector.

- Parameter form of a line: \(x = x_0 + ta, y = y_0 + tb, z = z_0 + tc\)
- You can obtain this simply by multiplying out the parameter form of the line, in which you find each component of the line \(\langle x, y, z\rangle\).
- The slopes are the direction vectors, and the \(y\)-intercepts are the position vectors

- Symmetric form of a line if \(a, b, c \neq 0\):

- Obtained by solving each equation for \(t\). \(t\) should be the same for points.
- Two lines are parallel if their direction vectors are parallel
- Two vectors intersect if they have a point in common
- Two lines are skew if they are not parallel and do not intersect.
- You can specify a certain \(t\) range.
- Planes - to find vector equations for all points on a plane.
- Normal vector - any vector that sticks out in a perpendicular function.
- Suppose we are given a normal vector (can be found using the cross product), a point \(R\), and an unknown point \(P(x, y, z)\).
- \(\vec{r} - \vec{r}_0\) is in the plane.
- \(\vec{r} - \vec{r}_0\) is orthogonal to \(\vec{n}\), the normal vector.
- Vector form: \((\vec{r} - \vec{r}_0) \cdot \vec{n} = 0\).

- If \(\vec{n} = \langle a, b, c \rangle, \vec{r} = \langle x, y, z \rangle, \vec{r}_0 = \langle x_0, y_0, z_0 \rangle\), then we have

This is the standard form/scalar equation of the plane.

If two planes are not parallel, they intersect along a line; the acute angle of intersection is the acute angle between their normal vectors.

\[\vec{n}_1 \cdot \vec{n}_2 = \| \vec{n}_1 \| \| \vec{n_2} \| \cos \theta\]## Week 2 Friday (12.6)

- Goal: to become familiarized with 7 three-dimensional shapes and names: cylinders, cones, ellipsoids, paraboloids (2 types), hyperboloids (2 types).
- Two-dimensional shape review
- Line
- Parabola: \(ax^2 + by = c\) or \(ax + by^2 = c\).
- Ellipse: \(ax^2 + by^2 = c\), \(a, b, c > 0\).
- Hyperbola: \(ax^2 - by^2 = c\) or \(-ax^2 + by^2 = c\).

- Cylinders
- If one variable is absent, then the graph is a two-dimensional curve extended into the third dimension.
- Circular cylinder: \(x^2 + y^2 = 1\)
- Cosine cylinder: \(z = \cos x\)

- Quadratic surfaces
- Any surface given by an equation involving sum of first and second powers of \(x, y, z\).
- \(ax^2 + bx + cy^2 + dy + ez^2 + fz = g\).
- To visualize, use traces. Fix one variable at a time and then find the resulting 2D shape. Fix the other two to find the cross-section and find the resulting plane.

- Elliptical/Circular Paraboloid
- \[\frac{x^2}{a^2} + \frac{y^2}{b^2} = \frac{z}{c}\]

- Hyperbolic Paraboloid
- \[\frac{x^2}{a^2} - \frac{y^2}{b^2} = \frac{z}{c}\]
- Traces build together a surface.

- Ellipsoid/Sphere
- \[\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1\]

- Circular/Elliptical Cone
- \[\frac{x^2}{a^2} + \frac{y^2}{eb2} = \frac{z^2}{c^2}\]

- Hyperboloid of one sheet
- \[\frac{x^2}{a^2} + \frac{b^2}{e^2} - \frac{z^2}{f^2}= 1\]

- Hyperboloid of two sheets
- \[\frac{x^2}{a^2} + \frac{b^2}{e^2} - \frac{z^2}{f^2}= 1\]

- In all of the above, replace \(\text{var}\) with \((\text{var} - r\) to shift the future.

## Week 3 Monday (13.1)

- Goals:
- Given curve equations, describe what the curve looks like
- Given surfaces, find curve equations for the intersections

- We allow for vector components to be expressions of an independent variable \(t\). You can read out the parametric equations component-wise.
- Space curve. A line is indeed a space curve with a specific term in which the \(t\) can be factored out.
- How does the space curve look like in the \(x, y, z\) coordinate systems? There is no \(t\) in the coordinate system - \(t\) is just a parameter, a time.
- Eliminate \(t\) from the equation and find the curve in the \(x, y, z\) coordinate system.
- Surface of motion - the surface upon which the particle moves. We must have one equation.
- When trigonometry is involved, invoke the Pythagorean theorem. In general algebra, try to use substitution to obtain only one equation involving \(x\) and/or \(y\) and/or \(z\).
- Intersection between a curve and a surface: plug in and solve for times, then plug back into the curve or the surface to obtain points.
- Intersection between two curves: do they intersect? Do they collide?
- Intersecting two surfaces. Begin by finding a parametrization for two surfaces.
- Let one of \(x, y, z\) be equal to \(t\) and solve the others in terms of \(t\).
- Set the most complicated independent variable to \(t\).

- For circles/ellipses, try \(x = a\cos t\) and \(y = b\sin t\).

- Let one of \(x, y, z\) be equal to \(t\) and solve the others in terms of \(t\).

## Week 3 Wednesday (13.2)

- Derivatives and integrals of vector functions.
- \(t\) can be interpreted as a time.
- Limit of a vector: take the limits of the individual components of a vector function.
- Derivative of the vector: take component-wise derivative.
- Continuous vector functions
- The unit tangent vector \(\vec{T}(t) = \frac{\vec{r}'t}{\| \vec{r}'(t) \|}\).
- The tangent vector cannot be a zero-vector.
- The tangent line is parallel to \(\vec{r}'(t)\) and passes through the point \(\vec{r}(t)\).
- The integral of a vector is done component-wise.
- The distance traveled on a curve from time \(a\) to \(b\) is:

- This is arc length if the arc is not traversed multiple times.

## Week 3 Friday (13.3a)

- Given a curve, we will assume that its derivative is always continuous.
- Continuity is considered component-wise.
- We can compute the distance along a curve by integrating the magnitude of the derivative.
- Arclength function: feed arc-length with a variable and make the arc length dependent on a variable - the endpoint \(t\).

- Fundamental theorem of analysis: \(f(x) = \frac{d}{dx} \int_a^x f(u) du\).
- \(\frac{s(t)}{dt}\) is the speed function - distance over time.
- Re-parametrization: describe the same curve with different equations.
- Parametrization in terms of \(s\): where is the vector location after it has traveled \(s\) units along the curve?
- \(\vec{T}'\) measures change in the direction of the curve, represented via \(\vec{T}\)>
- Curvature \(K\) or \(\kappa\) - measure of how quickly a curve changes direction at some point.

For two dimensional vectors, expand dimensions to the third dimension and simplify.

\(\frac{1}{\kappa(t)}44 is the radius of the curvature: it is the radius of the circle that best matches the curve, dependent on\)t$$.

## Week 4 Monday (13.3b)

- Unit tangent vector for a smooth curve - \(\vec{T}(t)\).
- Principal unit normal vector/normal vector - \(\vec{N}(t)\).

- \(\vec{T}(t)\) and \(\vec{T}'(t)\) are orthogonal.
- Binormal vector - \(\vec{B}(t)\)$

- We can create a time-dependent coordinate system with three orthogonal axes using these vectors. This is a TNB frame.

- \(\vec{B}(t)\) is a unit vector.
- Normal plane - plane spanned by \(\vec{N}(t)\) and \(\vec{B}(t)\).
- The normal plane is orthogonal to the unit tangent vector; thus, some point on a curve \(P\) and the derivative of the curve at \(P\) can be plugged into the plane equation to define the normal plane.

## Week 4 Friday (13.4)

- Velocity vector \(\vec{v}\). Speed \(s = \| \vec{v} \|\).
- Derivative of velocity - acceleration.

Problem.An object of mass 10kg is being acted on by a force \(\vec{F}(t) = \langle 130t, 10e^t, 10e^{-t} \rangle\). You are given \(\vec{v}(0) = \langle 0, 0, 1 \rangle\) and \(\vec{r}(0) = \langle 0, 1, 1 \rangle\). Find the position function \(\vec{r}(t)\).

Solution.If we integrate both sides, we have that \(\vec{a} = \vec{r}'' \implies \langle \frac{130}{2}t^2, 10e^t - 10, -10e^-t + 20 \rangle = \int 10 \cdot \vec{r}'(t)\). Integrating again gives \(\langle \frac{65}{3}t^3, 10e^t - 10t, 10e^{-t} + 20t \rangle = 10 \vec{r}(t)\). Dividing everything by 10 yields the curve.

- Reference to acceleration and velocity in that very moment.
- Two components: a tangent and normal component of acceleration.

Differentiating both sides:

\[\vec{T}' \nu + \vec{T} \nu' = \vec{v}'(t)\]Using the formula for curvature, we obtain

\[\kappa = \frac{\| \vec{T}' \|}{v} \implies \kappa \nu = \| \vec{T}' \|\] \[\vec{N} = \frac{\vec{T}'}{\| \vec{T}' \| \implies \vec{T}' = \kappa \nu \vec{N}\] \[\kappa \nu^2 \vec{N} + \nu' \vec{T} = \vec{a}(t)\]- We have written acceleration as a sum of \(\vec{T}\) and \(\vec{N}\).
- \[a_\vec{T} = \nu', a_\vec{N} = kappa \nu^2\]

## Week 5 Monday (14.1)

- Multivariable functions
- A real function \(f\) of two variables assigns each pair
- Domain: the set of all values that are allowed.
- Range: the set of all values the output takes on.
- Look for domain restrictions - square roots, logarithms, rationals, absolute values, etc.
- Intercepts are helpful (esp. when planes are involved).
- Use traces to generate level curves - contour map/elevation map as a projection of level curves.

## Week 5 Wednesday (14.7)

- Critical numbers - places where the critical number is zero or not defined in the domain.
- Secpmd derivative test - check concavity
- A local maximum occurs at \(9a, b)\) if \(f(a, b) \ge f(x, y)$4 for all points\)(x, y)\(near\)(a, b)$$. This is the local minimum value.
- Some extrema can happen at special locations. A critcial point is a point at which \(f_x(a, b) = 0\) and \(f_y(a, b)=0\), or \(f_x(a, b)44 or\)f_y(a, b)$$ do not exist.
- Saddle point - point which is not a minimum or maximum
- Second derivative test - directional derivatives.
- Compute \(D = f_{xx} (a, b) f_{yy}(a, b) - f_{xy}^2(a, b)\).
- If \(D > 044, concavity is the same in all directions. If\)f_{xx} > 0\(, it is a local minimum (concave up in all directions). If\)f_{xx} < 0$$, it is concave down in all directions, a local maximum.
- If \(D < 0\), saddle point.
- If \(D = 0\), inconclusive.

## Week 6 Wednesday (15.1)

- Double integral - signed volume between a function and the \(xy\) plane.
- Integrate across both dimensions.

## Week 6 Friday (15.2)

- Given \(x\) in the range \(a \le x \le b\), we have \(g_1(x) \le y \le g_2(x)\). Top/bottom. Compute integral as \(\int_a^b \left( \int_{g_1(x)}^{g_2(x)} f(x, y) dy \right)\)

## Week 7 Monday (10.3)

- Polar coordinates - stand at the origin facing the positive \(x\)-axis.
- Rotate by \(\theta\)
- Conversion: \(x = r \cos \theta\), \(y = r \sin theta\), \(\tan \theta = \frac{y}{x}\), \(x^2 + y^2 = r^2\)

## Week 7 Wednesday (15.3)

- In polar coordinates, \(x = r \cos \theta, y = r \sin \theta\)
- To set up a double integral, describe the region with polar coordinates, replace \(x\) and \(y\), and replace \(dA\) with \(r dr d \theta\).
- Why replace \(dA\) with \(r dr d \theta\)? The arc length is \(r d \theta\), the ‘length’ is \(dr\).

## Week 7 Friday (15.4)

- Mass and center of mass
- The density \(\rho(x, y)\) of some lamina varies with \(x, y\); \(\rho(x, y)\) is measured in units of mass p unit area.
- Mass of whole lamina: \(\int \int_D \rho (x, y) dA\)
- Inverse/proportionality: adapt a proportionality constant \(k\)
- Moment of the lamina about the \(x\)-axis: \(M_x = \int \int_D y \rho(x, y) dA\)
- The center of mass can lie outside of the lamina.

## Week 8 Wednesday (Taylor Notes 1)

- Taylor polynomial of degree 1: \(T_1(x) = f(b) + f'(b)(x - b)\)
- Approximation error
- Bounds - given a function \(f(x)\) - a number is an upper bound if all values are less than or equal to that value.
- You don’t need a ‘best’ bound, just
*a*bound that works. - Error boudns: given an interval \(I\) aroudn \(b\), \(b - \epsilon \le x \le b + \epsilon\): if \(\| f''(x) \| \le M\) for all \(x\) in \(I\), then

## Week 8 Friday (Taylor Notes 2)

- 1st Taylor Polynomial: the tangent line. The error is bounded by \(\frac{M}{2} \| x - b \|^2\) for \(\| f''(x) \| M\). You can make this independent of \(x\) by finding the maximum possible RHS expression on the domain.
- The second Taylor polynomial of a function \(f(x)\) based at \(b\) is

- This error will be less than or equal to \(\frac{M}{6} \| x - b \|^3\) for \(\| f'''(x) \| le M\).

The error for the \(n\)th Taylor polynomial is \(\frac{M}{(n+1)!} \| x-b \|^{n+1}\)

We are matching the derivative on multiple different levels.

## Week 9 Monday (Taylor Notes 3)

- Sigma notation - allows us to express complex patterns, like exponentials of \(-1\) to change sign
- \(n\)th Taylor polynomial - as \(n\) increases, the error bound decreases.
- We let \(n \to \infty\)

Taylor Series for \(f(x)\) at base \(b\):

\[\lim_{n\to\infty} T_n(x) = \sum_{k=0}^\infty \frac{1}{k!} f^k (b)(x-b)^k\]- If the limit exists for an \(x\), it converges at \(x\). Otherwise, it diverges at \(x\).
- Interval of convergence - largest open interval over which the series converges.
- Convergence only happens within an interval.
- Important series:
- \(e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} + ...\), converges for all \(x\)
- \(\sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} \pm ...\), converges for all \(x\)
- \(\cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} \pm ...\), converges for all \(x\)

## Week 9 Wednesday (Taylor Notes 4)

- \(n\)th Taylor polynomial of \(f\) on \(I\): \(T_n(x) = \sum_{k=0}^n \frac{1}{k!} f^k (b) (x-b)^k\)
- Error bound: error \(\lke \frac{M}{(n+1)!} \|x-b\|^{n+1}\)
- Taylor series - take \(n\to\infty\), we see error \(\to 0\).
- Word of caution - sometimes a function is differentiable and \(T_n\) doesn’t diverge, but it does not work out.
- Interval of convergence: when do we see convergence? Which \(x\) leads to convergence when \(n\to\infty\)?
- Consider \((1 - x) T_n(x) = ...\), slve for \(T_n (x)\).

## Week 9 Friday (Taylor Series 5)

- Developing additional ability to write Taylor series
- On the interval of convergence, we can pull in constants and merge Taylor series.
- Whenever we have an integral/derivative and a sum, we can distribute across it term-wise.

- Same logic can be applied to the integral
- The differentiation/integration will have the same interval of convergence as the original

## Week 10 Wednesday

- Take derivatives repeatedly and find patterns in evaluation at certain points.
- For \(f(x) = \sum_{k=0}^\infty _k (x - b)^k\), \(f^n b = c_n \cdot n!\), \(c_n = \frac{f^n (b)}{n!}\)