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MATH 200: What to Memorize Before the Exam

Only the essentials: the things you can't work out quickly in an exam. Practice them as flashcards.

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Vectors, Dot Product & Cross Product

  • a⋅b=a1b1+a2b2+a3b3=∣a∣ ∣b∣cos⁡θ\mathbf a\cdot\mathbf b = a_1b_1 + a_2b_2 + a_3b_3 = |\mathbf a|\,|\mathbf b|\cos\theta
  • a⊥b=a⋅b=0\mathbf a \perp \mathbf b = \mathbf a\cdot\mathbf b = 0
  • projab\text{proj}_{\mathbf a}\mathbf b
    a⋅b∣a∣2 a\dfrac{\mathbf a\cdot\mathbf b}{|\mathbf a|^2}\,\mathbf a
    The is a⋅b∣a∣\dfrac{\mathbf a\cdot\mathbf b}{|\mathbf a|}.
  • a×b=⟨a2b3−a3b2, a3b1−a1b3, a1b2−a2b1⟩\mathbf a\times\mathbf b = \langle a_2b_3 - a_3b_2,\ a_3b_1 - a_1b_3,\ a_1b_2 - a_2b_1\rangle
    Perpendicular to both. Watch the middle component's order.
  • ∣a×b∣=∣a∣ ∣b∣sin⁡θ|\mathbf a\times\mathbf b| = |\mathbf a|\,|\mathbf b|\sin\theta
    The parallelogram's area. Halve it for a triangle.
  • Volume of a parallelepiped\text{Volume of a parallelepiped}
    ∣a⋅(b×c)∣|\mathbf a\cdot(\mathbf b\times\mathbf c)|
  • x2+y2+z2=R2=sphere of radius Rx^2 + y^2 + z^2 = R^2 = \text{sphere of radius } R
    Centered at (x0,y0,z0)(x_0, y_0, z_0): (x−x0)2+(y−y0)2+(z−z0)2=R2(x - x_0)^2 + (y - y_0)^2 + (z - z_0)^2 = R^2. With linear terms, complete the square to find the center.

Equations of Lines in 2D and 3D

  • Line\text{Line}
    ⟨x,y,z⟩=P+t d\langle x, y, z\rangle = P + t\,\mathbf d
    When every di≠0d_i \ne 0, the line through (x0,y0,z0)(x_0, y_0, z_0) has the symmetric form x−x0d1=y−y0d2=z−z0d3\dfrac{x - x_0}{d_1} = \dfrac{y - y_0}{d_2} = \dfrac{z - z_0}{d_3}.
  • Direction of the line where two planes meet\text{Direction of the line where two planes meet}
    n1×n2\mathbf n_1\times\mathbf n_2
  • Distance from Q to the line P+t d\text{Distance from } Q \text{ to the line } P + t\,\mathbf d
    ∣(Q−P)×d∣∣d∣\dfrac{|(Q - P)\times\mathbf d|}{|\mathbf d|}

Equations of Planes

  • Plane\text{Plane}
    a(x−x0)+b(y−y0)+c(z−z0)=0a(x - x_0) + b(y - y_0) + c(z - z_0) = 0
    ⟨a,b,c⟩\langle a, b, c\rangle is the .
  • Distance from (x0,y0,z0) to ax+by+cz=d\text{Distance from } (x_0, y_0, z_0) \text{ to } ax + by + cz = d
    ∣ax0+by0+cz0−d∣a2+b2+c2\dfrac{|ax_0 + by_0 + cz_0 - d|}{\sqrt{a^2 + b^2 + c^2}}

Space Curves, Tangent Vectors & Arc Length

  • Velocity, speed\text{Velocity, speed}
    r′(t),∣r′(t)∣\mathbf r'(t),\quad |\mathbf r'(t)|
  • Arc length\text{Arc length}
    ∫ab∣r′(t)∣ dt\int_a^b |\mathbf r'(t)|\,dt

Sketching Surfaces & Level Curves

  • x2+y2+z2=R2=sphere of radius Rx^2 + y^2 + z^2 = R^2 = \text{sphere of radius } R
    Centered at (x0,y0,z0)(x_0, y_0, z_0): (x−x0)2+(y−y0)2+(z−z0)2=R2(x - x_0)^2 + (y - y_0)^2 + (z - z_0)^2 = R^2. With linear terms, complete the square to find the center.
  • x2a2+y2b2+z2c2=1=ellipsoid\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} + \dfrac{z^2}{c^2} = 1 = \text{ellipsoid}
  • z=x2+y2=paraboloidz = x^2 + y^2 = \text{paraboloid}
    The variable that is not squared gives the axis.
  • z2=x2+y2=conez^2 = x^2 + y^2 = \text{cone}
  • x2+y2−z2=1=hyperboloid of one sheetx^2 + y^2 - z^2 = 1 = \text{hyperboloid of one sheet}
    A circle at every height.
  • x2+y2−z2=−1=hyperboloid of two sheetsx^2 + y^2 - z^2 = -1 = \text{hyperboloid of two sheets}
    No points for ∣z∣<1|z| < 1.
  • x2+y2=1=cylinder along the z-axisx^2 + y^2 = 1 = \text{cylinder along the } z\text{-axis}
    A missing variable means every value of it is allowed.

Limits & Continuity in Two Variables

  • x, y=rcos⁡θ, rsin⁡θx,\ y = r\cos\theta,\ r\sin\theta
  • x2+y2=r2x^2 + y^2 = r^2

Partial Derivatives

  • fx(a,b)=lim⁡h→0f(a+h,b)−f(a,b)hf_x(a, b) = \lim_{h \to 0}\dfrac{f(a + h, b) - f(a, b)}{h}
  • fxy=fyxf_{xy} = f_{yx}
    True when both are continuous. This is Clairaut's theorem.
  • ∂z∂x for F(x,y,z)=0=−FxFz\dfrac{\partial z}{\partial x} \text{ for } F(x, y, z) = 0 = -\dfrac{F_x}{F_z}
    Only where Fz≠0F_z \ne 0.

The Multivariable Chain Rule

  • ddt f(x(t),y(t))=fx dxdt+fy dydt\dfrac{d}{dt}\,f\big(x(t), y(t)\big) = f_x\,\dfrac{dx}{dt} + f_y\,\dfrac{dy}{dt}
    With more variables, write one term per path in the .

Tangent Planes & Normal Lines

  • Tangent plane to z=f(x,y) at (a,b)\text{Tangent plane to } z = f(x, y) \text{ at } (a, b)
    z=f(a,b)+fx(a,b)(x−a)+fy(a,b)(y−b)z = f(a, b) + f_x(a, b)(x - a) + f_y(a, b)(y - b)
    The right side is the linear approximation L(x,y)L(x, y).
  • Normal to z=f(x,y)\text{Normal to } z = f(x, y)
    ⟨fx, fy, −1⟩\langle f_x,\ f_y,\ -1\rangle
  • Normal to F(x,y,z)=c\text{Normal to } F(x, y, z) = c
    ∇F(x0,y0,z0)\nabla F(x_0, y_0, z_0)
    Plane: ∇F⋅⟨x−x0,y−y0,z−z0⟩=0\nabla F \cdot \langle x - x_0, y - y_0, z - z_0\rangle = 0.
  • Tangent to F=c, G=k\text{Tangent to } F = c,\ G = k
    ∇F×∇G\nabla F \times \nabla G
    Direction of the curve where the two surfaces meet.

Linear Approximation & Error

  • Tangent plane to z=f(x,y) at (a,b)\text{Tangent plane to } z = f(x, y) \text{ at } (a, b)
    z=f(a,b)+fx(a,b)(x−a)+fy(a,b)(y−b)z = f(a, b) + f_x(a, b)(x - a) + f_y(a, b)(y - b)
    The right side is the linear approximation L(x,y)L(x, y).
  • Δf=≈fx Δx+fy Δy\Delta f = \approx f_x\,\Delta x + f_y\,\Delta y
    Maximum error: use ∣fx∣ Δx+∣fy∣ Δy|f_x|\,\Delta x + |f_y|\,\Delta y.

Directional Derivatives & the Gradient

  • ∇f=⟨fx, fy, fz⟩\nabla f = \langle f_x,\ f_y,\ f_z\rangle
  • Duf=∇f⋅uD_{\mathbf u}f = \nabla f \cdot \mathbf u
    u\mathbf u must be a .
  • Max rate of change\text{Max rate of change}
    ∣∇f∣|\nabla f|
    In the direction of ∇f\nabla f. Fastest decrease: −∇f-\nabla f, rate −∣∇f∣-|\nabla f|.

Critical Points & the Second Derivative Test

  • D=fxxfyy−fxy2D = f_{xx}f_{yy} - f_{xy}^2
  • D>0, fxx>0=local minD > 0,\ f_{xx} > 0 = \text{local min}
  • D>0, fxx<0=local maxD > 0,\ f_{xx} < 0 = \text{local max}
  • D<0=saddle pointD < 0 = \text{saddle point}
  • D=0=no conclusionD = 0 = \text{no conclusion}
    Look at ff itself near the point.

Absolute Maximum & Minimum on a Region

  • Absolute max, min on a closed region\text{Absolute max, min on a closed region}
    critical points inside, boundary pieces, corners\text{critical points inside, boundary pieces, corners}
    Evaluate ff at every candidate and compare.

Lagrange Multipliers

  • Extremes of f on g=k\text{Extremes of } f \text{ on } g = k
    ∇f=λ∇g,g=k\nabla f = \lambda\nabla g,\quad g = k
  • On g=k1 and h=k2\text{On } g = k_1 \text{ and } h = k_2
    ∇f=λ∇g+μ∇h\nabla f = \lambda\nabla g + \mu\nabla h
  • Closest point to (x0,y0,z0)\text{Closest point to } (x_0, y_0, z_0)
    minimize (x−x0)2+(y−y0)2+(z−z0)2\text{minimize } (x - x_0)^2 + (y - y_0)^2 + (z - z_0)^2
    Gives the same point as the distance, without the square root.

Double Integrals & Order of Integration

  • Volume between two surfaces\text{Volume between two surfaces}
    ∬D(top−bottom) dA\iint_D (\text{top} - \text{bottom})\,dA

Double Integrals in Polar Coordinates

  • x, y=rcos⁡θ, rsin⁡θx,\ y = r\cos\theta,\ r\sin\theta
  • x2+y2=r2x^2 + y^2 = r^2
  • dA=r dr dθdA = r\,dr\,d\theta
    Never drop the rr.
  • x2+y2=2ax=r=2acos⁡θx^2 + y^2 = 2ax = r = 2a\cos\theta
    For a>0a > 0, −π2≤θ≤π2-\tfrac{\pi}{2} \le \theta \le \tfrac{\pi}{2}. And x2+y2=2ayx^2 + y^2 = 2ay is r=2asin⁡θr = 2a\sin\theta, 0≤θ≤π0 \le \theta \le \pi.

Mass, Center of Mass & Surface Area

  • M=∬Dρ dAor∭Eρ dVM = \iint_D \rho\,dA \quad \text{or} \quad \iiint_E \rho\,dV
    Here ρ\rho is the , not the spherical ρ\rho. With constant density, MM is the density times the area or volume.
  • xˉ=1M∬Dx ρ dA\bar x = \dfrac{1}{M}\iint_D x\,\rho\,dA
    Same for yˉ\bar y with yρy\rho. For a solid, zˉ=1M∭Ezρ dV\bar z = \frac{1}{M}\iiint_E z\rho\,dV.
  • favg=1A(D)∬Df dAf_{\text{avg}} = \dfrac{1}{A(D)}\iint_D f\,dA
  • area of z=f(x,y)\text{area of } z = f(x, y)
    ∬D1+fx2+fy2 dA\iint_D \sqrt{1 + f_x^2 + f_y^2}\,dA
  • ∬x2+y2≤a2a2−x2−y2 dA=23πa3\iint_{x^2 + y^2 \le a^2} \sqrt{a^2 - x^2 - y^2}\,dA = \tfrac{2}{3}\pi a^3
    This is the volume of a half ball. Use the geometry formula instead of integrating.

Triple Integrals

  • M=∬Dρ dAor∭Eρ dVM = \iint_D \rho\,dA \quad \text{or} \quad \iiint_E \rho\,dV
    Here ρ\rho is the , not the spherical ρ\rho. With constant density, MM is the density times the area or volume.
  • xˉ=1M∬Dx ρ dA\bar x = \dfrac{1}{M}\iint_D x\,\rho\,dA
    Same for yˉ\bar y with yρy\rho. For a solid, zˉ=1M∭Ezρ dV\bar z = \frac{1}{M}\iiint_E z\rho\,dV.

Triple Integrals in Cylindrical Coordinates

  • x2+y2=2ax=r=2acos⁡θx^2 + y^2 = 2ax = r = 2a\cos\theta
    For a>0a > 0, −π2≤θ≤π2-\tfrac{\pi}{2} \le \theta \le \tfrac{\pi}{2}. And x2+y2=2ayx^2 + y^2 = 2ay is r=2asin⁡θr = 2a\sin\theta, 0≤θ≤π0 \le \theta \le \pi.
  • dV (cylindrical)=r dz dr dθdV\ \text{(cylindrical)} = r\,dz\,dr\,d\theta
    With x=rcos⁡θx = r\cos\theta, y=rsin⁡θy = r\sin\theta, z=zz = z.

Triple Integrals in Spherical Coordinates

  • (x, y, z)=(ρsin⁡φcos⁡θ, ρsin⁡φsin⁡θ, ρcos⁡φ)(x,\ y,\ z) = (\rho\sin\varphi\cos\theta,\ \rho\sin\varphi\sin\theta,\ \rho\cos\varphi)
    Here r=ρsin⁡φr = \rho\sin\varphi is the cylindrical rr.
  • x2+y2+z2=ρ2x^2 + y^2 + z^2 = \rho^2
  • dV (spherical)=ρ2sin⁡φ dρ dφ dθdV\ \text{(spherical)} = \rho^2\sin\varphi\,d\rho\,d\varphi\,d\theta
  • z=x2+y2=φ=π4z = \sqrt{x^2 + y^2} = \varphi = \tfrac{\pi}{4}
    In general, for k>0k > 0, z=kx2+y2z = k\sqrt{x^2 + y^2} is tan⁡φ=1k\tan\varphi = \tfrac{1}{k}.
  • x2+y2+(z−a)2=a2=ρ=2acos⁡φx^2 + y^2 + (z - a)^2 = a^2 = \rho = 2a\cos\varphi
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