APMAE2000 Multivariable

APMAE2000 Multivariable Difficult Problem Memo Multivariable Exam Preparation

• I don’t quite understand why the calculations are different between the Divergence Theorem and Green’s Theorem.
  • Why is the flux in Green’s Theorem the divergence in the Divergence Theorem?

• I finally understood the meaning after combining the two (blu3mo).
  • The difference between $d\vec{S}$ and $dS$, for example.
  • After converting to $dA$, I was finally able to do the double integral of u, v.

• In the first place, when partial derivatives cannot be calculated, it is not smooth.

• Is this equivalent to the existence of a normal vector?

• Surface area:

  • Taking the cross product of tangent vectors gives the area of a small parallelogram.

• Surface integral:

  • Similar to area, it is obtained by multiplying the length of the normal vector (= area of the parallelogram).
  • In the case of a vector integral, the orientation of the normal vector also matters.
    • Orientable: It means that the front and back can be defined.
    • In 3D, are there surfaces that are not orientable?
      • Like a Möbius strip.
      • In this case, vector surface integral cannot be defined.

Green’s Theorem:

• I completely understood it in this video.

  • If you do a double integral of curl, it becomes exactly a line integral.

  • To break it down further,

      - I want to check this (blu3mo).
    

  • I should teach curl first.

Conservative Fields:

• A vector that always points in the direction of maximum slope of the potential function at that position.
• Why is it called conservative?
  • No matter which route you take for the line integral, the value is the same. This means that energy is conserved.
  • To verify this, we use $f_{xy}=f_{yx}$.
  • I want to connect this to the understanding of physics (blu3mo)(blu3mo).
• Calculation to create f from F:
  • g(y) is found to eliminate terms when taking $\partial F/\partial x$.

Line Integral:

• When integrating arc length, you can also multiply it by the value of f(r(t)).
  • Independence of parameterization should be confirmed, as it often appears in multiple-choice questions.
  • Even for vector fields, you can add up the components in the direction of r’ by multiplying the unit vector of r’.
• For example, when integrating with respect to dx in the range of θ from 0 to pi, you multiply by x’(t)=-sinθ.
  • The reason for multiplying by negative makes sense because you are integrating in the opposite direction of x.
• The transformation from F dr to F * T ds should be understood by solving problems.
  • T is the tangent vector of r, not F.
• Hw9 Q3 is useful, so it should be solved.
  • It provides insights into keeping r simple.
    • Well, but if the range is not 0 to n, the subsequent calculations become difficult.
    • It is important to gain intuition about that.

Vector Field:

• It’s self-evident.
• Understanding that the potential of F (vector -> vector) is f (vector -> scalar) is important (blu3mo).

Double Integral:

• 
• 
  • The inner integral with respect to dy, the expression of x is only in terms of y.

Lagrange Multipliers:

• When there is f(x, y)=~~

Topology:

• 
• All points are interior => Open Set
• All boundary points are included -> Closed Set
  • The definition of open set is different (blu3mo).
• Theorems:
  • A local extremum of f occurs at an interior point P => P is a critical point.
  • Extreme Value Theorem: If the set D is closed and bounded, $f:D\to R$ and R is continuous, global maximum/minimum exists?
  • TODO: It seems obvious, so I want to find counterexamples outside the conditions (blu3mo).

20221018:

• 
• Terms:
  • Maximum: f(x)
    • NOT x itself
  • Maximizer: x
• Definition: critical point
  • When:
    • Case 1: The function is not differentiable.
      • $f_x(x_0,y_0)$ or $f_y(x_0,y_0)$ does not exist.
    • Case 2: ∇f(x)=0, x is a critical point.
    • There are two conditions.
    • For example, a local max/min is a critical point.
      • But there are also other cases.
  • Saddle point: Not a local max/min but ∇f(x)=0.
    • Similar to an inflection point.
    • But it can exist in places where it is a maximum in the x-direction but a minimum in the y-direction.
  • It is necessary to verify max/min/saddle point.- Use the formula $D=f_{xx}{f}_{yy}-(f_{xy}{)}^2$.
• I understand the conditional branching, but I can’t visualize the reason.

2022101?

• https://openstax.org/books/calculus-volume-3/pages/4-6-directional-derivatives-and-the-gradient

• I skipped the class, but reading this helped me understand it intuitively.

• The gradient ∇f is a vector given by $f_x(x,y)\hat{i}+f_y(x,y)\hat{j}$.

  • The direction of this vector is the direction of maximum slope.

• u is a vector given by $\cos \theta \hat{i}+\sin \theta \hat{j}$.

• $\nabla f\cdot u$ is the slope in the direction of u.

  • The maximum value of $\nabla f\cdot u$ in a certain direction is achieved when the direction of u coincides with the direction of ∇f, which means $\nabla f\cdot u=|\nabla f||u|\cos (\Phi )$ is maximized when $\Phi =0$.
  • It is important to understand that these three conditions are equivalent.

• ∇f

• Theorem 2. The gradient vector ∇f has the following properties:

• 1. The gradient is orthogonal to level sets.

  • This also holds for a three-dimensional function f(x,y,z).
    • The normal vector to the level set plane is obtained.

• 2. It points in the direction of greatest change.

• 3. Its magnitude is the amount of greatest change.

20221013

• Differential
  • I still don’t have an intuition for why we only add the x-direction and y-direction.
• Chain Rule
  • This also adds the x-direction and y-direction partial derivatives.
• Chain Rule with two variables
  • Same as above.

20221011

• Differentiability

  • When we want to know if f(x) is differentiable at x=p:
    • At that point, if we can express f(x)-L(x)=E(x)(x-p), then we can say f(x) is differentiable.
      • Here, as x approaches p, E(x) approaches 0.
      • L(x) is a linear function, like $a_0{x}_0+a_1{x}_1+...+k$.
    • In other words, in Japanese, if we can approximate f(x) with a linear function and the error approaches 0 as x approaches p, then f(x) is differentiable.
      • Not writing the error=0 at x=p because we don’t know if that value exists.
  • When we do this with vectors:
    • Basically the same idea, we calculate the tangent plane and see if the value exists as x approaches p.
    • L(x) becomes the tangent plane, I see.
    • It is important to understand the problem of determining whether it is differentiable.- Scalar Fields = Functions of multiple variables
  • Instead of range, it is called image.
  • 
  • The domain can be obtained by transforming the constraints if there are square roots or other restrictions in the function.
  • The image is the set of all possible output values of the function.

• Understanding Form

  • Imagine setting f(x) = z and manipulating the equation to understand the form it takes when z is a certain value.
    • This method can also be used to draw contour lines.
    • elispc

• Limit

  • The value of a limit can change depending on the direction in which x and y approach.

    • Check the “cardinal” directions:

    • That is, check limx→a f(x,b) and limy→b f(a,y). If they don’t match or either doesn’t

    • exist, the limit as a whole does not exist.

  • 
  • There are five methods to find limits:
    • Plug-in
      • The simplest method.
    • Cardinal
      • This method can be used to check if a limit exists.
    • Other directions
      • Substitute y=0, y=x, y=x^2, etc. to check if the limit exists.
      • Since there are infinitely many directions, it is not possible to prove the existence of a limit by trying all of them.
    • Squeezing
      • Requires knowledge of the inequality |2xy|≦x^2+y^2.
    • Polar
      • Important point to remember: Changing variables from (x,y) to (0,0) corresponds to r→0.
        • It is convenient because there is only one variable.
    • Tips
      • It is important to make sure the denominator is not zero, the numerator can be dealt with later.
  • I don’t really understand limits well (blu3mo)
    • I don’t understand why ${\lim }_{x\to 0}f(x)=0$ when $f(x)=x$.
      • Ah, I understand now.
      • Even though x≠0, isn’t f(x) still equal to 0?
        • It can be understood by looking at the definition.

20220922

• Arc Length
  • Is it about line integrals?
  • Can we just take the length every time we do a vector integral?
    • Is it like ${\int }_b^a|\vec{r}(t)|dt$? (blu3mo)
    • No, the vector |r(t)| is not the arc length, but the distance from the origin.
  • It should be something like ${\int }_b^a|\frac{\vec{r}(t)-\vec{r}(t+dt)}{dt}|dt$ (blu3mo)
    • Then, it can actually be simplified to ${\int }_b^a|{\vec{r}}^{\prime }(t)|dt$.
    • Indeed, r’(t) is the tangent vector, so adding them up should work (blu3mo)
  • When calculating, expand |r’(t)| as $\sqrt{x^2+y^2}$ and integrate.
    • It’s tedious (blu3mo)
• Arc-Length Parameterization
  • This allows us to find the position after moving a certain distance.
• Curvature
  • The curvature measures how smoothly the position changes with respect to the second derivative.
  • However, it is important to note that $\kappa =|\frac{d}{ds}T(s)|$.
    • The parameter s is obtained from the arc-length parameterization, not t.
    • T is the unit tangent vector.
  • But this formula involves s and makes calculations more complicated, so there is a simpler formula.
    • $\kappa =|\frac{T^{\prime }(t)}{r^{\prime }(t)}|$
  • The proof that the curvature of a circle is 1/R should be followed (blu3mo)
    • 
    • I did it, if I can do it from scratch from 0, it should be fine.
  • There are also some formulas that are recommended to memorize:
    • Helix: The curvature of $(acost,asint,bt)$ is constant and equal to $\frac{a}{(a^2+b^2)}$.
      • Is it about spirals?
  • Understanding r(s) is also important

ℝ20220920

• Vector Integral
  • ${\int }_b^a\vec{r}(t)dt=({\int }_b^{a}x(t)dt,{\int }_b^{a}y(t)dt,{\int }_b^{a}z(t)dt)$
    • Just like differentiation, integration is done separately for each dimension.
• Specific Application: Mechanics
  • $\vec{v}(t)=\vec{v}(t_0)+{\int }_{t_0}^{t}a(t^{\prime })d{t}^{\prime }$
  • $\vec{x}(t)=\vec{x}(t_0)+{\int }_{t_0}^{t}v(t^{\prime })d{t}^{\prime }$
  • In the end, even if it becomes a vector, each dimension is calculated independently.
    • It’s like solving separate problems in two dimensions.

20220915

• Distance Calculation for Line/Plane and Point/Plane
  • Previously, we calculated the distance by dropping a perpendicular from a point to a line or plane.- However, this is a manipulation close to proj, so it can be calculated using proj.
• Distance between a point $\vec{x}$ and a line $l=\vec{p}+\lambda \vec{v}$:
  • $d=(\vec{x}-\vec{p})-pro{j}_{\vec{v}}(\vec{x}-\vec{p})$
  • The difference between $\vec{x}-\vec{p}$ and its projection onto the direction vector is the distance, I see. (blu3mo)
• Distance between a point $\vec{x}$ and a plane $(\vec{r}-\vec{p})\cdot \vec{n}=0$:
  • It’s simpler. $com{p}_{\vec{n}}(\vec{x}-\vec{p})=|pro{j}_{\vec{n}}(\vec{x}-\vec{p})|$
    • By subtracting the position vector and then projecting onto the normal vector, we can obtain the distance.
    • Since the normal vector grows from the position vector, it is necessary to subtract the position vector.
• Shortest distance between two skewed lines:
  • Line $l=\vec{p}+\lambda \vec{v},m=\vec{q}+\omega \vec{r}$
  • First, take the normal of the two line directions.
    • $\vec{n}=\vec{v}\times \vec{r}$
  • Then, project $\vec{p}-\vec{q}$ onto the normal to obtain the shortest distance.
    • Here, the distance information is not meaningful with respect to the normal, but the difference between p and q is meaningful. (blu3mo)

---

• Vector limits and derivatives
  • Consider a parametric curve.
    • $\vec{v}=(f(t),g(t),h(t))$
    • (If f, g, h are linear functions, it becomes a line.)
  • When we differentiate,
    • $r^{\prime }(x)={\lim }_{h\to 0}\frac{\vec{r}(x+h)-\vec{r}(x)}{h}$
    • This is just a single variable differentiation, except that the function is a vector.
  • Here, chain rule and product rule also work for vector differentiation.
    • It works for dot product and cross product.
      • Oh, why? (blu3mo)
      • https://openstax.org/books/calculus-volume-3/pages/3-2-calculus-of-vector-valued-functions
        • There was a proof, but I can’t intuitively understand it.
  • I need to review this. (blu3mo)
  • The unit tangent vector is defined as $\frac{{\vec{r}}^{\prime }(t)}{|r^{\prime }(t)|}$.
    • Yes, that makes sense. (blu3mo)
    • In the case of the derivative of a scalar x, y is on the left side, but in the case of three dimensions, both x and y are on the right side, so taking the derivative gives the tangent.
    • 
      • What’s the difference between left and right?
      • Simply put, in the left example, f(x) is included in the y vector of the right example.

20220913

• Definition of a line
  • Known information
  • When considering whether two lines are parallel, there are various ways to think about it.
    • Whether $\vec{a}=\lambda \vec{b}$ or not
    • Whether $|\vec{a}\cdot \vec{b}|=|\vec{a}||\vec{b}|$ or not (a and b are directional vectors)
      • This is the same as $\vec{a}=\lambda \vec{b}$, right..?
    • Whether there are two intersection points or not
• Planes
  • Known information
  • The set of points where the dot product of the normal vector and (position vector - x) is 0.

---

• proj/comp

  • I understood it for now, and even if I forget it, I can understand it quickly by reading something like this: https://web.ma.utexas.edu/users/m408m/Display12-3-4.shtml
  • When it comes to $pro{j}_{x}a$, x only has directional information (distance information doesn’t matter), so it’s important to be aware of that.
    • It makes sense when you think about it, and even when you calculate it, the distance information disappears because it is divided by the distance.

• dot/cross product

  • Most of it is already known.

  • Be careful because they are not commutative and not linear.

    • 

  • $|\vec{a}\times \vec{b}|$ gives the area of the parallelogram formed by a and b, and also gives the area of the parallelepiped.

---

• Textbook, either one is fine.

• hw, completion + correctness

• We will study multivariable calculus.

  • As the name suggests.

• Locus

  • A set of points defined by an equation.

• Vector

  • It only has the difference between two coordinates (we don’t know which is the starting and ending point).
  • Magnitude: $|\vec{V}|=\sqrt{{v_1}^2+{v_2}^2+{v_3}^2}$
  • Linear combination, probably linear combination
    • $a\vec{v}+b\vec{v}$
    • Scalar multiplication and vector addition are necessary to define this (the definition is trivial).

• Thoughts

  • It seems to be difficult in the future, but not as difficult as Foundations of Mathematical Sciences.