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?

image image

  • I finally understood the meaning after combining the two (blu3mo).
    • The difference between and , for example.
    • After converting to , I was finally able to do the double integral of u, v.

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  • 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 .
    • 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 .

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:

  • image
  • image
    • 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:

  • image
  • 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, and R is continuous, global maximum/minimum exists?
    • TODO: It seems obvious, so I want to find counterexamples outside the conditions (blu3mo).

20221018:

  • image
  • Terms:
    • Maximum: f(x)
      • NOT x itself
    • Maximizer: x
  • Definition: critical point
    • When:
      • Case 1: The function is not differentiable.
        • or 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 .
  • 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 .

    • The direction of this vector is the direction of maximum slope.
  • u is a vector given by .

  • is the slope in the direction of u.

    • The maximum value of in a certain direction is achieved when the direction of u coincides with the direction of ∇f, which means is maximized when .
    • 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.
    1. It points in the direction of greatest change.
    1. 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 .
      • 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.
    • 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.

    • image
    • 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 when .
        • 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 ? (blu3mo)
      • No, the vector |r(t)| is not the arc length, but the distance from the origin.
    • It should be something like (blu3mo)
      • Then, it can actually be simplified to .
      • Indeed, r’(t) is the tangent vector, so adding them up should work (blu3mo)
    • When calculating, expand |r’(t)| as 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 .
      • 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.
    • The proof that the curvature of a circle is 1/R should be followed (blu3mo)
      • image
      • 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 is constant and equal to .
        • Is it about spirals?
    • Understanding r(s) is also important

ℝ20220920

  • Vector Integral
      • Just like differentiation, integration is done separately for each dimension.
  • Specific Application: Mechanics
    • 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 and a line :
    • The difference between and its projection onto the direction vector is the distance, I see. (blu3mo)
  • Distance between a point and a plane :
    • It’s simpler.
      • 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
    • First, take the normal of the two line directions.
    • Then, project 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.
      • (If f, g, h are linear functions, it becomes a line.)
    • When we differentiate,
      • 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.
    • I need to review this. (blu3mo)
    • The unit tangent vector is defined as .
      • 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.
      • image
        • 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 or not
      • Whether or not (a and b are directional vectors)
        • This is the same as , 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 , 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.

      • image
    • 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:
    • Linear combination, probably linear combination
      • Scalar multiplication and vector addition are necessary to define this (the definition is trivial).
  • Thoughts