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 $dS$ 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 $∂F/∂x$.
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 multiplechoice 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.
 It provides insights into keeping r simple.
Vector Field:
 It’s selfevident.
 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, $f:D→R$ 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
 Maximum: f(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.
 Case 1: The function is not differentiable.
 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 xdirection but a minimum in the ydirection.
 It is necessary to verify max/min/saddle point. Use the formula $D=f_{xx}f_{yy}−(f_{xy})_{2}$.
 When:
 I understand the conditional branching, but I can’t visualize the reason.
2022101?

https://openstax.org/books/calculusvolume3/pages/46directionalderivativesandthegradient

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

The gradient ∇f is a vector given by $f_{x}(x,y)i^+f_{y}(x,y)j^ $.
 The direction of this vector is the direction of maximum slope.

u is a vector given by $cosθi^+sinθj^ $.

$∇f⋅u$ is the slope in the direction of u.
 The maximum value of $∇f⋅u$ in a certain direction is achieved when the direction of u coincides with the direction of ∇f, which means $∇f⋅u=∣∇f∣∣u∣cos(Φ)$ is maximized when $Φ=0$.
 It is important to understand that these three conditions are equivalent.

∇f

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

 The gradient is orthogonal to level sets.
 This also holds for a threedimensional function f(x,y,z).
 The normal vector to the level set plane is obtained.

 It points in the direction of greatest change.

 Its magnitude is the amount of greatest change.
20221013
 Differential
 I still don’t have an intuition for why we only add the xdirection and ydirection.
 Chain Rule
 This also adds the xdirection and ydirection 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)(xp), 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.
 At that point, if we can express f(x)L(x)=E(x)(xp), then we can say f(x) is differentiable.
 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.
 When we want to know if f(x) is differentiable at x=p:

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
 Imagine setting f(x) = z and manipulating the equation to understand the form it takes when z is a certain value.

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:
 Plugin
 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.
 Important point to remember: Changing variables from (x,y) to (0,0) corresponds to r→0.
 Tips
 It is important to make sure the denominator is not zero, the numerator can be dealt with later.
 Plugin
 I don’t really understand limits well (blu3mo)
 I don’t understand why $lim_{x→0}f(x)=0$ 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.
 I don’t understand why $lim_{x→0}f(x)=0$ when $f(x)=x$.
 The value of a limit can change depending on the direction in which x and y approach.
20220922
 Arc Length
 Is it about line integrals?
 Can we just take the length every time we do a vector integral?
 Is it like $∫_{b}∣r(t)∣dt$? (blu3mo)
 No, the vector r(t) is not the arc length, but the distance from the origin.
 It should be something like $∫_{b}∣dtr(t)−r(t+dt) ∣dt$ (blu3mo)
 Then, it can actually be simplified to $∫_{b}∣r_{′}(t)∣dt$.
 Indeed, r’(t) is the tangent vector, so adding them up should work (blu3mo)
 When calculating, expand r’(t) as $x_{2}+y_{2} $ and integrate.
 It’s tedious (blu3mo)
 ArcLength 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 $κ=∣dsd T(s)∣$.
 The parameter s is obtained from the arclength 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.
 $κ=∣r_{′}(t)T_{′}(t) ∣$
 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 $(a_{2}+b_{2})a $.
 Is it about spirals?
 Helix: The curvature of $(acost,asint,bt)$ is constant and equal to $(a_{2}+b_{2})a $.
 Understanding r(s) is also important
ℝ20220920
 Vector Integral
 $∫_{b}r(t)dt=(∫_{b}x(t)dt,∫_{b}y(t)dt,∫_{b}z(t)dt)$
 Just like differentiation, integration is done separately for each dimension.
 $∫_{b}r(t)dt=(∫_{b}x(t)dt,∫_{b}y(t)dt,∫_{b}z(t)dt)$
 Specific Application: Mechanics
 $v(t)=v(t_{0})+∫_{t_{0}}a(t_{′})dt_{′}$
 $x(t)=x(t_{0})+∫_{t_{0}}v(t_{′})dt_{′}$
 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 $x$ and a line $l=p +λv$:
 $d=(x−p )−proj_{v}(x−p )$
 The difference between $x−p $ and its projection onto the direction vector is the distance, I see. (blu3mo)
 Distance between a point $x$ and a plane $(r−p )⋅n=0$:
 It’s simpler. $comp_{n}(x−p )=∣proj_{n}(x−p )∣$
 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.
 It’s simpler. $comp_{n}(x−p )=∣proj_{n}(x−p )∣$
 Shortest distance between two skewed lines:
 Line $l=p +λv,m=q +ωr$
 First, take the normal of the two line directions.
 $n=v×r$
 Then, project $p −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.
 $v=(f(t),g(t),h(t))$
 (If f, g, h are linear functions, it becomes a line.)
 When we differentiate,
 $r_{′}(x)=lim_{h→0}hr(x+h)−r(x) $
 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/calculusvolume3/pages/32calculusofvectorvaluedfunctions
 There was a proof, but I can’t intuitively understand it.
 It works for dot product and cross product.
 I need to review this. (blu3mo)
 The unit tangent vector is defined as $∣r_{′}(t)∣r_{′}(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.
 Consider a parametric curve.
20220913
 Definition of a line
 Known information
 When considering whether two lines are parallel, there are various ways to think about it.
 Whether $a=λb$ or not
 Whether $∣a⋅b∣=∣a∣∣b∣$ or not (a and b are directional vectors)
 This is the same as $a=λb$, 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/Display1234.shtml
 When it comes to $proj_{x}a$, 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.

$∣a×b∣$ 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: $∣V∣=v_{1}_{2}+v_{2}_{2}+v_{3}_{2} $
 Linear combination, probably linear combination
 $av+bv$
 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.