Link Search Menu Expand Document

Preclass Notes

MATH 126

Table of contents
  1. Week 1 Monday (12.1)
  2. Week 1 Wednesday (12.2)
  3. Week 1 Friday (12.3)
  4. Week 2 Monday (12.4, 12.5)
  5. Week 2 Wednesday (12.5)
  6. Week 2 Friday (12.6)
  7. Week 3 Monday (13.1)
  8. Week 3 Wednesday (13.2)
  9. Week 3 Friday (13.3a)
  10. Week 4 Monday (13.3b)
  11. Week 4 Friday (13.4)
  12. Week 5 Monday (14.1)
  13. Week 5 Wednesday (14.7)
  14. Week 6 Wednesday (15.1)
  15. Week 6 Friday (15.2)
  16. Week 7 Monday (10.3)
  17. Week 7 Wednesday (15.3)
  18. Week 7 Friday (15.4)
  19. Week 8 Wednesday (Taylor Notes 1)
  20. Week 8 Friday (Taylor Notes 2)
  21. Week 9 Monday (Taylor Notes 3)
  22. Week 9 Wednesday (Taylor Notes 4)
  23. Week 9 Friday (Taylor Series 5)
  24. 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:
\[\mathbb{R} \times \mathbb{R} \times \mathbb{R} = \mathbb{R}^3\] \[= {(x, y, z) | x, y, z \in \mathbb{R}\]
  • 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.

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 \(a, b, c \neq 0\):
\[\frac{x - x_a}{a} = \frac{y - y_0}{b} = \frac{z - z_0}{c}\]
  • 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
\[a(x - x_0) + b(y - y_0) + c(z - z_0) = 0\]

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:
    1. Given curve equations, describe what the curve looks like
    2. 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 \(y = b\sin t\).

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:
\[\int_a^b \| \vec{r}'(t) \| dt = \int_a^b \sqrt{x'(t)^2 + y'(t)^2 + z'(t)^2} dt\]
  • 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 \(t\).
\[s(t) = \int_a^b \| \vec{r}'(u) \| du\]
  • 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.
\[\kappa = \| \frac{d\vec{T}}{ds} \| \approx \| \frac{\text{change in direction}}{\text{change in distance}} \|\] \[K(t) = \frac{\| \vec{r}'(t) \times \vec{r}''(t) \|}{ \| \vec{r}'(t) \|^3} = \frac{\| \vec{T}'(t)}{\| \vec{r}'(t) \|}\]
  • 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 - \(\vec{N}(t)\).
\[\vec{N}(t) = \frac{\vec{T}'(t)}{\| \vec{T} '(t) \| }\]
  • \(\vec{T}(t)\) and \(\vec{T}'(t)\) are orthogonal.
  • Binormal vector - \(\vec{B}(t)\)$
\[\vec{B}(t) = \vec{T}(t) \times \vec{N}(t)\]
  • We can create a time-dependent coordinate system with three orthogonal axes using these vectors. This is a TNB frame.
\[\| \vec{B}(t) \| = \| \vec{T}(t) \| \cdot \| \vec{N} (t) \| \cdot \sin \theta \implies 1 = 1 \cdot 1 \cdot 1\]
  • \(\vec{B}(t)\) 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.
\[\vec{F} = m \cdot \vec{a}\]

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.
\[\nu = \text{speed}\] \[\vec{T}(t) = \frac{\vec{v}(t)}{\nu} \implies \vec{T} \nu = \vec{v}(t)\]

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
\[\| f(x) - T_1(x) \| \le \frac{M}{2} \| x - b \|^2\]

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
\[T_2(x) = T_1(x) + \frac{1}{2} f''(b)(x-b)^2\]
  • This error will be less than or equal to \(\frac{M}{6} \| x - b \|^3\) for \(\| f'''(x) \| le M\).
\[T_3(x) = T_2(x) + \frac{1}{6} f'''(b)(x-b)^3\] \[T_n(x) = T_{n-1}(x) + \frac{1}{n!} f^n(b)(x-b)^n\]
  • 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.
\[\frac{d}{dx} A(x) = \sum_{k=0}^\infty \frac{d}{dx} \left( \frac{f^k (b)}{k!} (x - b)^k \right) = \sum_{k = 0}^\infty \frac{f^k (b)}{(k-1)!} (x - b)^{k-1}\]
  • 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 \(f(x) = \sum_{k=0}^\infty _k (x - b)^k\), \(f^n b = c_n \cdot n!\), \(c_n = \frac{f^n (b)}{n!}\)