Vectors and Geometry of Space
Description
Equations
Distance formula
d = ( x 1 − x 2 ) 2 + ( y 1 − y 2 ) 2 + ( z 1 − z 2 ) 2 d = \sqrt{(x_1-x_2)^2 + (y_1-y_2)^2 + (z_1-z_2)^2} d = ( x 1 − x 2 ) 2 + ( y 1 − y 2 ) 2 + ( z 1 − z 2 ) 2
Equation of a sphere ( a , b , c ) (a,b,c) ( a , b , c )
( x − a ) 2 + ( y − b ) 2 + ( z − c ) 2 = r 2 (x-a)^2 + (y-b)^2 + (z-c)^2 = r^2 ( x − a ) 2 + ( y − b ) 2 + ( z − c ) 2 = r 2
Norm/length/magnitude of vectors
∣ a ∣ = a 1 2 + a 2 2 + a 3 2 \lvert \mathbf{a} \rvert = \sqrt{a_1^2 + a_2^2 + a_3^2} ∣ a ∣ = a 1 2 + a 2 2 + a 3 2
Vector addition and subtraction
a ± b = ⟨ a 1 ± b 1 , a 2 ± b 2 , a 3 ± b 3 ⟩ \mathbf{a} \pm \mathbf{b} = \langle a_1 \pm b_1, a_2 \pm b_2, a_3 \pm b_3 \rangle a ± b = ⟨ a 1 ± b 1 , a 2 ± b 2 , a 3 ± b 3 ⟩
Vector scalar multiplication
c a = ⟨ c a 1 , c a 2 , c a 3 ⟩ c\mathbf{a} = \langle ca_1, ca_2, ca_3 \rangle c a = ⟨ c a 1 , c a 2 , c a 3 ⟩
Properties of vectors
a + b = b + a a + ( b + c ) = ( a + b ) + c a + 0 = a a + ( − a ) = 0 c ( a + b ) = c a + c b ( c + d ) a = c a + d a ( c d ) a = c ( d a ) 1 a = a \mathbf{a}+\mathbf{b} = \mathbf{b}+\mathbf{a} \newline \mathbf{a}+(\mathbf{b}+\mathbf{c}) = (\mathbf{a}+\mathbf{b})+\mathbf{c} \newline \mathbf{a}+\mathbf{0}=\mathbf{a} \newline \mathbf{a}+(-\mathbf{a}) = \mathbf{0} \newline c(\mathbf{a}+\mathbf{b}) = c\mathbf{a}+c\mathbf{b} \newline (c+d)\mathbf{a} = c\mathbf{a}+d\mathbf{a} \newline (cd)\mathbf{a} = c(d\mathbf{a}) \newline 1\mathbf{a}=\mathbf{a} a + b = b + a a + ( b + c ) = ( a + b ) + c a + 0 = a a + ( − a ) = 0 c ( a + b ) = c a + c b ( c + d ) a = c a + d a ( c d ) a = c ( d a ) 1 a = a
Standard basis vectors
i = ⟨ 1 , 0 , 0 ⟩ j = ⟨ 0 , 1 , 0 ⟩ k = ⟨ 0 , 0 , 1 ⟩ \mathbf{i} = \langle 1,0,0 \rangle \newline \mathbf{j} = \langle 0,1,0 \rangle \newline \mathbf{k} = \langle 0,0,1 \rangle i = ⟨ 1 , 0 , 0 ⟩ j = ⟨ 0 , 1 , 0 ⟩ k = ⟨ 0 , 0 , 1 ⟩
Description
Equations
Dot product
a ⋅ b = a 1 b 1 + a 2 b 2 + a 3 b 3 \mathbf{a}\cdot\mathbf{b} = a_1b_1 + a_2b_2 + a_3b_3 a ⋅ b = a 1 b 1 + a 2 b 2 + a 3 b 3
Properties of dot product
a ⋅ a = ∣ a ∣ 2 a ⋅ b = b ⋅ a a ⋅ ( b + c ) = a ⋅ b + a ⋅ c ( c a ) ⋅ b = c ( a ⋅ b ) = a ⋅ ( c b ) 0 ⋅ a = 0 \mathbf{a}\cdot\mathbf{a}=\lvert\mathbf{a}\rvert^2 \newline \mathbf{a}\cdot\mathbf{b}=\mathbf{b}\cdot\mathbf{a} \newline \mathbf{a}\cdot(\mathbf{b}+\mathbf{c}) = \mathbf{a}\cdot\mathbf{b} + \mathbf{a}\cdot\mathbf{c} \newline (c\mathbf{a})\cdot\mathbf{b}=c(\mathbf{a}\cdot\mathbf{b})=\mathbf{a}\cdot(c\mathbf{b}) \newline \mathbf{0}\cdot\mathbf{a} = \mathbf{0} a ⋅ a = ∣ a ∣ 2 a ⋅ b = b ⋅ a a ⋅ ( b + c ) = a ⋅ b + a ⋅ c ( c a ) ⋅ b = c ( a ⋅ b ) = a ⋅ ( c b ) 0 ⋅ a = 0
Dot product and angle between vectors
a ⋅ b = ∣ a ∣ ∣ b ∣ cos θ cos θ = a ⋅ b ∣ a ∣ ∣ b ∣ \mathbf{a}\cdot\mathbf{b} = \lvert\mathbf{a}\rvert\lvert\mathbf{b}\rvert\cos\theta \newline \cos\theta = \dfrac{\mathbf{a}\cdot\mathbf{b}}{\lvert\mathbf{a}\rvert\lvert\mathbf{b}\rvert} a ⋅ b = ∣ a ∣ ∣ b ∣ cos θ cos θ = ∣ a ∣ ∣ b ∣ a ⋅ b
Dot product to check orthogonal vectors
a ⋅ b = 0 \mathbf{a}\cdot\mathbf{b} = 0 a ⋅ b = 0
Direction angles and direction cosines
cos α = a ⋅ i ∣ a ∣ ∣ i ∣ = a 1 ∣ a ∣ cos β = a 2 ∣ a ∣ cos γ = a 3 ∣ a ∣ \cos\alpha = \dfrac{\mathbf{a}\cdot\mathbf{i}}{\lvert\mathbf{a}\rvert\lvert\mathbf{i}\rvert} = \dfrac{a_1}{\lvert\mathbf{a}\rvert} \newline \cos\beta = \dfrac{a_2}{\lvert\mathbf{a}\rvert} \newline \cos\gamma = \dfrac{a_3}{\lvert\mathbf{a}\rvert} cos α = ∣ a ∣ ∣ i ∣ a ⋅ i = ∣ a ∣ a 1 cos β = ∣ a ∣ a 2 cos γ = ∣ a ∣ a 3
Unit vector and direction cosines
a ∣ a ∣ = ⟨ cos α , cos β , cos γ ⟩ \dfrac{\mathbf{a}}{\lvert\mathbf{a}\rvert} = \langle \cos\alpha, \cos\beta, \cos\gamma \rangle ∣ a ∣ a = ⟨ cos α , cos β , cos γ ⟩
Scalar projection of b \mathbf{b} b a \mathbf{a} a
c o m p a b = a ⋅ b ∣ a ∣ \mathrm{comp}_{\mathbf{a}}\mathbf{b} = \dfrac{\mathbf{a}\cdot\mathbf{b}}{\lvert\mathbf{a}\rvert} c o m p a b = ∣ a ∣ a ⋅ b
Vector projection of b \mathbf{b} b a \mathbf{a} a
p r o j a b = a ⋅ b ∣ a ∣ a ∣ a ∣ = a ⋅ b ∣ a ∣ 2 a \mathrm{proj}_{\mathbf{a}}\mathbf{b} = \dfrac{\mathbf{a}\cdot\mathbf{b}}{\lvert\mathbf{a}\rvert}\dfrac{\mathbf{a}}{\lvert\mathbf{a}\rvert} = \dfrac{\mathbf{a}\cdot\mathbf{b}}{\lvert\mathbf{a}\rvert^2}\mathbf{a} p r o j a b = ∣ a ∣ a ⋅ b ∣ a ∣ a = ∣ a ∣ 2 a ⋅ b a
Description
Equations
Cross product
a × b = ∣ a 1 a 2 a 3 b 1 b 2 b 3 c 1 c 2 c 3 ∣ = ⟨ a 2 b 3 − a 3 b 2 , a 3 b 1 − a 1 b 3 , a 1 b 2 − a 2 b 1 ⟩ \mathbf{a}\times\mathbf{b} = \begin{vmatrix} a_1 & a_2 & a_3 \cr b_1 & b_2 & b_3 \cr c_1 & c_2 & c_3 \end{vmatrix} = \newline \langle a_2b_3-a_3b_2, a_3b_1-a_1b_3, a_1b_2-a_2b_1\rangle a × b = ∣ ∣ ∣ ∣ ∣ ∣ ∣ a 1 b 1 c 1 a 2 b 2 c 2 a 3 b 3 c 3 ∣ ∣ ∣ ∣ ∣ ∣ ∣ = ⟨ a 2 b 3 − a 3 b 2 , a 3 b 1 − a 1 b 3 , a 1 b 2 − a 2 b 1 ⟩
Cross product to generate orthogonal vectors
a × b \mathbf{a}\times\mathbf{b} a × b a \mathbf{a} a b \mathbf{b} b
Cross product and angle between vectors
∣ a × b ∣ = ∣ a ∣ ∣ b ∣ sin θ \lvert\mathbf{a}\times\mathbf{b}\rvert = \lvert\mathbf{a}\rvert\lvert\mathbf{b}\rvert \sin\theta ∣ a × b ∣ = ∣ a ∣ ∣ b ∣ sin θ
Cross product to check parallel vectors
a × b = 0 \mathbf{a}\times\mathbf{b} = \mathbf{0} a × b = 0
Cross product as the area of parallelogram
A = ∣ a × b ∣ A = \lvert\mathbf{a}\times\mathbf{b}\rvert A = ∣ a × b ∣
Cross products of standard basis vectors
i × j = k j × k = i k × i = j \mathbf{i}\times\mathbf{j} = \mathbf{k} \newline \mathbf{j}\times\mathbf{k} = \mathbf{i} \newline \mathbf{k}\times\mathbf{i} = \mathbf{j} i × j = k j × k = i k × i = j
Properties of cross product
a × b = − b × a ( c a ) × b = c ( a × b ) = a × ( c b ) a × ( b + c ) = a × b + a × c ( a + b ) × c = a × c + b × c a ⋅ ( b × c ) = ( a × b ) ⋅ c a × ( b × c ) = ( a ⋅ c ) b − ( a ⋅ b ) c \mathbf{a}\times\mathbf{b} = -\mathbf{b}\times\mathbf{a} \newline (c\mathbf{a})\times\mathbf{b} = c(\mathbf{a}\times\mathbf{b}) = \mathbf{a}\times(c\mathbf{b}) \newline \mathbf{a}\times(\mathbf{b}+\mathbf{c}) = \mathbf{a}\times\mathbf{b}+\mathbf{a}\times\mathbf{c} \newline (\mathbf{a}+\mathbf{b})\times\mathbf{c} = \mathbf{a}\times\mathbf{c} + \mathbf{b}\times\mathbf{c} \newline \mathbf{a}\cdot(\mathbf{b}\times\mathbf{c}) = (\mathbf{a}\times\mathbf{b})\cdot\mathbf{c} \newline \mathbf{a}\times(\mathbf{b}\times\mathbf{c}) = (\mathbf{a}\cdot\mathbf{c})\mathbf{b} - (\mathbf{a}\cdot\mathbf{b})\mathbf{c} a × b = − b × a ( c a ) × b = c ( a × b ) = a × ( c b ) a × ( b + c ) = a × b + a × c ( a + b ) × c = a × c + b × c a ⋅ ( b × c ) = ( a × b ) ⋅ c a × ( b × c ) = ( a ⋅ c ) b − ( a ⋅ b ) c
Scalar triple product as volume of parallelepiped
V = ∣ a ⋅ ( b × c ) ∣ V = \lvert \mathbf{a}\cdot(\mathbf{b}\times\mathbf{c}) \rvert V = ∣ a ⋅ ( b × c ) ∣
Scalar triple product to check three coplanar vectors
V = a ⋅ ( b × c ) = 0 V=\mathbf{a}\cdot(\mathbf{b}\times\mathbf{c})=0 V = a ⋅ ( b × c ) = 0
Description
Equations
Vector equation of a line
r = r 0 + t v \mathbf{r} = \mathbf{r}_0 + t\mathbf{v} r = r 0 + t v
Parametric equations of a line ( x 0 , y 0 , z 0 ) (x_0, y_0, z_0) ( x 0 , y 0 , z 0 ) ⟨ a , b , c ⟩ \langle a, b, c \rangle ⟨ a , b , c ⟩
x = x 0 + a t y = y 0 + b t z = z 0 + c t x=x_0+at \newline y = y_0+bt \newline z=z_0+ct x = x 0 + a t y = y 0 + b t z = z 0 + c t
Symmetric equation of a line
x − x 0 a = y − y 0 b = z − z 0 c \dfrac{x-x_0}{a} = \dfrac{y-y_0}{b} = \dfrac{z-z_0}{c} a x − x 0 = b y − y 0 = c z − z 0
Description
Equations
Vector equation of a line segment
r ( t ) = ( 1 − t ) r 0 + t r 1 t ∈ [ 0 , 1 ] \mathbf{r}(t) = (1-t)\mathbf{r}_0+t\mathbf{r}_1 \newline t\in[0,1] r ( t ) = ( 1 − t ) r 0 + t r 1 t ∈ [ 0 , 1 ]
Vector equation of a plane
n ⋅ ( r − r 0 ) = 0 n ⋅ r = n ⋅ r 0 \mathbf{n}\cdot(\mathbf{r}-\mathbf{r}_0) = 0 \newline \mathbf{n}\cdot\mathbf{r} = \mathbf{n}\cdot\mathbf{r}_0 n ⋅ ( r − r 0 ) = 0 n ⋅ r = n ⋅ r 0
Scalar equation of a plane ( x 0 , y 0 , z 0 ) (x_0, y_0, z_0) ( x 0 , y 0 , z 0 ) ⟨ a , b , c ⟩ \langle a,b,c \rangle ⟨ a , b , c ⟩
a ( x − x 0 ) + b ( y − y 0 ) + c ( z − z 0 ) = 0 a(x-x_0)+b(y-y_0)+c(z-z_0)=0 a ( x − x 0 ) + b ( y − y 0 ) + c ( z − z 0 ) = 0
Linear equation of a plane
a x + b y + c z + d = 0 ax+by+cz+d=0 a x + b y + c z + d = 0
Distance from a point to a plane
D = ∣ a x 1 + b y 1 + c z 1 + d ∣ a 2 + b 2 + c 2 D = \dfrac{\lvert ax_1+by_1+cz_1+d \rvert}{\sqrt{a^2+b^2+c^2}} D = a 2 + b 2 + c 2 ∣ a x 1 + b y 1 + c z 1 + d ∣
Cylinders and quadratic surfaces
Description
Equations
Ellipsoid
x 2 a 2 + y 2 b 2 + z 2 c 2 = 1 \dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}+\dfrac{z^2}{c^2}=1 a 2 x 2 + b 2 y 2 + c 2 z 2 = 1
Cone
z 2 c 2 = x 2 a 2 + y 2 b 2 \dfrac{z^2}{c^2}=\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2} c 2 z 2 = a 2 x 2 + b 2 y 2
Elliptic paraboloid
z c = x 2 a 2 + y 2 b 2 \dfrac{z}{c}=\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2} c z = a 2 x 2 + b 2 y 2
Hyperbolic paraboloid
z c = x 2 a 2 − y 2 b 2 \dfrac{z}{c}=\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2} c z = a 2 x 2 − b 2 y 2
Hyperboloid of one sheet
x 2 a 2 + y 2 b 2 − z 2 c 2 = 1 \dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}-\dfrac{z^2}{c^2}=1 a 2 x 2 + b 2 y 2 − c 2 z 2 = 1
Hyperboloid of two sheets
− x 2 a 2 − y 2 b 2 + z 2 c 2 = 1 -\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}+\dfrac{z^2}{c^2}=1 − a 2 x 2 − b 2 y 2 + c 2 z 2 = 1
Vector functions and space curves
Description
Equations
Vector-valued function
r ( t ) = ⟨ f ( t ) , g ( t ) , h ( t ) ⟩ \mathbf{r}(t) = \langle f(t), g(t), h(t) \rangle r ( t ) = ⟨ f ( t ) , g ( t ) , h ( t ) ⟩
Limit of a vector function
lim t → a r ( t ) = ⟨ lim t → a f ( t ) , lim t → a g ( t ) , lim t → a h ( t ) ⟩ \lim\limits_{t\to a}\mathbf{r}(t) = \langle \lim\limits_{t\to a}f(t), \lim\limits_{t\to a}g(t), \lim\limits_{t\to a}h(t) \rangle t → a lim r ( t ) = ⟨ t → a lim f ( t ) , t → a lim g ( t ) , t → a lim h ( t ) ⟩
Continuity of vector function
lim t → a r ( t ) = r ( t ) \lim\limits_{t\to a}\mathbf{r}(t) = \mathbf{r}(t) t → a lim r ( t ) = r ( t )
Parametric equation of space curves
x = f ( t ) y = g ( t ) z = h ( t ) x = f(t) \newline y = g(t) \newline z = h(t) x = f ( t ) y = g ( t ) z = h ( t )
Derivative of vector function
r ′ ( t ) = lim h → 0 r ( t + h ) − r ( t ) h \mathbf{r}'(t) = \lim\limits_{h\to 0}\dfrac{\mathbf{r}(t+h) - \mathbf{r}(t)}{h} r ′ ( t ) = h → 0 lim h r ( t + h ) − r ( t )
Derivative of vector function
r ′ ( t ) = ⟨ f ′ ( t ) , g ′ ( t ) , h ′ ( t ) ⟩ \mathbf{r}'(t) = \langle f'(t), g'(t), h'(t) \rangle r ′ ( t ) = ⟨ f ′ ( t ) , g ′ ( t ) , h ′ ( t ) ⟩
Differentiation rules
[ u ( t ) + v ( t ) ] ′ = u ′ ( t ) + v ′ ( t ) [\mathbf{u}(t)+\mathbf{v}(t)]' = \mathbf{u}'(t) + \mathbf{v}'(t)\newline [ u ( t ) + v ( t ) ] ′ = u ′ ( t ) + v ′ ( t ) [ c u ( t ) ] ′ = c u ′ ( t ) [c\mathbf{u}(t)]' = c\mathbf{u}'(t)\newline [ c u ( t ) ] ′ = c u ′ ( t ) [ f ( t ) u ( t ) ] ′ = f ′ ( t ) u ( t ) + f ( t ) u ′ ( t ) [f(t)\mathbf{u}(t)]' = f'(t)\mathbf{u}(t) + f(t)\mathbf{u}'(t)\newline [ f ( t ) u ( t ) ] ′ = f ′ ( t ) u ( t ) + f ( t ) u ′ ( t ) [ u ( t ) ⋅ v ( t ) ] ′ = u ′ ( t ) ⋅ v ( t ) + u ( t ) ⋅ v ′ ( t ) [\mathbf{u}(t)\cdot\mathbf{v}(t)]' = \mathbf{u}'(t)\cdot\mathbf{v}(t) + \mathbf{u}(t)\cdot\mathbf{v}'(t)\newline [ u ( t ) ⋅ v ( t ) ] ′ = u ′ ( t ) ⋅ v ( t ) + u ( t ) ⋅ v ′ ( t ) [ u ( t ) × v ( t ) ] ′ = u ′ ( t ) × v ( t ) + u ( t ) × v ′ ( t ) [\mathbf{u}(t)\times\mathbf{v}(t)]' = \mathbf{u}'(t)\times\mathbf{v}(t) + \mathbf{u}(t)\times\mathbf{v}'(t)\newline [ u ( t ) × v ( t ) ] ′ = u ′ ( t ) × v ( t ) + u ( t ) × v ′ ( t ) [ u ( f ( t ) ) ] ′ = f ′ ( t ) u ′ ( f ( t ) ) [\mathbf{u}(f(t))]' = f'(t)\mathbf{u}'(f(t)) [ u ( f ( t ) ) ] ′ = f ′ ( t ) u ′ ( f ( t ) )
Definite integral of vector function
∫ a b r ( t ) d t = ⟨ ∫ a b f ( t ) d t , ∫ a b g ( t ) d t , ∫ a b h ( t ) d t ⟩ \int_a^b \mathbf{r}(t) \ dt \newline = \langle \int_a^b f(t) \ dt, \int_a^b g(t) \ dt, \int_a^b h(t) \ dt \rangle ∫ a b r ( t ) d t = ⟨ ∫ a b f ( t ) d t , ∫ a b g ( t ) d t , ∫ a b h ( t ) d t ⟩
Position vector
r ( t ) \mathbf{r}(t) r ( t )
Tangent (velocity) vector
r ′ ( t ) \mathbf{r}'(t) r ′ ( t )
Unit tangent vector
T ( t ) = r ′ ( t ) ∣ r ′ ( t ) ∣ \mathbf{T}(t) = \dfrac{\mathbf{r}'(t)}{\lvert\mathbf{r}'(t)\rvert} T ( t ) = ∣ r ′ ( t ) ∣ r ′ ( t )
Arc length and curvature
Description
Equations
Length of a curve
L = ∫ a b ∣ r ′ ( t ) ∣ d t = ∫ a b [ f ( t ) ] 2 + [ g ( t ) ] 2 + [ h ( t ) ] 2 d t \begin{aligned}L &= \textstyle\int_a^b \lvert\mathrm{r}'(t)\rvert \ dt \cr &= \textstyle\int_a^b \sqrt{[f(t)]^2+[g(t)]^2+[h(t)]^2} \ dt\end{aligned} L = ∫ a b ∣ r ′ ( t ) ∣ d t = ∫ a b [ f ( t ) ] 2 + [ g ( t ) ] 2 + [ h ( t ) ] 2 d t
Arc length function
s ( t ) = ∫ a t ∣ r ′ ( u ) ∣ d u = ∫ a t [ f ( u ) ] 2 + [ g ( u ) ] 2 + [ h ( u ) ] 2 d u \begin{aligned}s(t) &= \textstyle\int_a^t \lvert\mathrm{r}'(u)\rvert \ du \cr &= \textstyle\int_a^t \sqrt{[f(u)]^2+[g(u)]^2+[h(u)]^2} \ du\end{aligned} s ( t ) = ∫ a t ∣ r ′ ( u ) ∣ d u = ∫ a t [ f ( u ) ] 2 + [ g ( u ) ] 2 + [ h ( u ) ] 2 d u
Rate of change in arc length and the tangent vector
d s d t = ∣ r ′ ( t ) ∣ \dfrac{ds}{dt} = \lvert \mathbf{r}'(t) \rvert d t d s = ∣ r ′ ( t ) ∣
Curvature
κ ( t ) = ∣ d T d s ∣ = ∣ T ′ ( t ) ∣ ∣ r ′ ( t ) ∣ = ∣ r ′ ( t ) × r ′ ′ ( t ) ∣ ∣ r ′ ( t ) ∣ 3 \kappa(t) = \bigg\lvert\dfrac{d\mathbf{T}}{ds}\bigg\rvert = \dfrac{\lvert\mathbf{T}'(t)\rvert}{\lvert\mathbf{r}'(t)\rvert} = \dfrac{\lvert\mathbf{r}'(t)\times\mathbf{r}''(t)\rvert}{\lvert\mathbf{r}'(t)\rvert^3} κ ( t ) = ∣ ∣ ∣ ∣ ∣ d s d T ∣ ∣ ∣ ∣ ∣ = ∣ r ′ ( t ) ∣ ∣ T ′ ( t ) ∣ = ∣ r ′ ( t ) ∣ 3 ∣ r ′ ( t ) × r ′ ′ ( t ) ∣
Curvature in terms of function
κ ( x ) = ∣ f ′ ′ ( x ) ∣ [ 1 + ( f ′ ( x ) ) 2 ] 3 / 2 \kappa(x) = \dfrac{\lvert f''(x)\rvert}{[1+(f'(x))^2]^{3/2}} κ ( x ) = [ 1 + ( f ′ ( x ) ) 2 ] 3 / 2 ∣ f ′ ′ ( x ) ∣
Unit normal vector
N ( t ) = T ′ ( t ) ∣ T ′ ( t ) ∣ \mathbf{N}(t) = \dfrac{\mathbf{T}'(t)}{\lvert\mathbf{T}'(t)\rvert} N ( t ) = ∣ T ′ ( t ) ∣ T ′ ( t )
Binormal vector
B ( t ) = T ( t ) × N ( t ) \mathbf{B}(t) = \mathbf{T}(t)\times\mathbf{N}(t) B ( t ) = T ( t ) × N ( t )
Radius of osculating circle
r = 1 κ r = \frac{1}{\kappa} r = κ 1
Velocity and acceleration
Description
Equations
Position vector
r ( t ) \mathbf{r}(t) r ( t )
Tangent (velocity) vector
v ( t ) = r ′ ( t ) \mathbf{v}(t) = \mathbf{r}'(t) v ( t ) = r ′ ( t )
Acceleration vector
a ( t ) = v ′ ( t ) \mathbf{a}(t) = \mathbf{v}'(t) a ( t ) = v ′ ( t )
Tangential and normal components of acceleration
a ( t ) = v ′ T + κ v 2 N \mathbf{a}(t) = v'\mathbf{T}+\kappa v^2\mathbf{N} a ( t ) = v ′ T + κ v 2 N
Description
Equations
Functions of two variables
f ( x , y ) ∣ ( x , y ) ∈ D z = f ( x , y ) {f(x,y) \vert (x, y) \in D} \newline z = f(x,y) f ( x , y ) ∣ ( x , y ) ∈ D z = f ( x , y )
Level curves
f ( x , y ) = k f(x,y) = k f ( x , y ) = k
Function of three variables
f ( x , y , z ) ∣ ( x , y , z ) ∈ E {f(x,y,z) \vert (x, y, z) \in E} f ( x , y , z ) ∣ ( x , y , z ) ∈ E
Level surfaces
f ( x , y , z ) = k f(x,y,z)=k f ( x , y , z ) = k
Function of n n n
f ( x ) = c ⋅ x f(\mathbf{x}) = \mathbf{c}\cdot\mathbf{x} f ( x ) = c ⋅ x
Interpretation of input of functions of several variables
1. n n n x 1 , x 2 , … , x n x_1, x_2,…, x_n x 1 , x 2 , … , x n ( x 1 , x 2 , … , x n ) (x_1, x_2,…, x_n) ( x 1 , x 2 , … , x n ) x = ⟨ x 1 , x 2 , … , x n ⟩ \mathbf{x} = \langle x_1, x_2,…, x_n \rangle x = ⟨ x 1 , x 2 , … , x n ⟩
Limit of function of two variables
lim ( x , y ) → ( a , b ) f ( x , y ) = L \lim\limits_{(x,y)\to(a,b)}f(x,y)=L ( x , y ) → ( a , b ) lim f ( x , y ) = L
Continuity of function of two variables
lim ( x , y ) → ( a , b ) f ( x , y ) = f ( a , b ) \lim\limits_{(x,y)\to(a,b)}f(x,y)=f(a,b) ( x , y ) → ( a , b ) lim f ( x , y ) = f ( a , b )
Limit of function of several variables
lim x → a f ( x ) = L \lim\limits_{\mathbf{x}\to\mathbf{a}}f(\mathbf{x})=L x → a lim f ( x ) = L
Continuity of function of several variables
lim x → a f ( x ) = f ( a ) \lim\limits_{\mathbf{x}\to\mathbf{a}}f(\mathbf{x})=f(\mathbf{a}) x → a lim f ( x ) = f ( a )
Description
Equations
Partial derivative with respect to x x x
f x ( x , y ) = lim h → 0 f ( x + h , y ) − f ( x , y ) h f_x(x,y) = \lim\limits_{h\to 0}\dfrac{f(x+h,y)-f(x,y)}{h} f x ( x , y ) = h → 0 lim h f ( x + h , y ) − f ( x , y )
Partial derivative with respect to y y y
f y ( x , y ) = lim h → 0 f ( x , y + h ) − f ( x , y ) h f_y(x,y) = \lim\limits_{h\to 0}\dfrac{f(x,y+h)-f(x,y)}{h} f y ( x , y ) = h → 0 lim h f ( x , y + h ) − f ( x , y )
Partial derivative rule
1. To find f x f_x f x y y y f ( x , y ) f(x,y) f ( x , y ) x x x f y f_y f y x x x f ( x , y ) f(x,y) f ( x , y ) y y y
Clairaut’s theorem
f x y ( a , b ) = f y x ( a , b ) f_{xy}(a,b) = f_{yx}(a,b) f x y ( a , b ) = f y x ( a , b )
Tangent plane and linear approximations
Description
Equations
Tangent plane to a surface z = f ( x , y ) z=f(x,y) z = f ( x , y ) ( x 0 , y 0 , z 0 ) (x_0,y_0,z_0) ( x 0 , y 0 , z 0 )
z − z 0 = f x ( x 0 , y 0 ) + f y ( x 0 , y 0 ) ( y − y 0 ) z-z_0=f_x(x_0,y_0)+f_y(x_0,y_0)(y-y_0) z − z 0 = f x ( x 0 , y 0 ) + f y ( x 0 , y 0 ) ( y − y 0 )
Linear approximation
f ( x , y ) ≈ f ( a , b ) + f x ( a , b ) ( x − a ) + f y ( a , b ) ( y − b ) f(x,y) \approx f(a,b)+f_x(a,b)(x-a)+f_y(a,b)(y-b) f ( x , y ) ≈ f ( a , b ) + f x ( a , b ) ( x − a ) + f y ( a , b ) ( y − b )
Total differential
d z = f x ( x , y ) d x + f y ( x , y ) d y dz = f_x(x,y) dx + f_y(x,y) dy d z = f x ( x , y ) d x + f y ( x , y ) d y
Description
Equations
Critical point
a point with f x ( a , b ) = 0 f_x(a,b)=0 f x ( a , b ) = 0 f y ( a , b ) = 0 f_y(a,b)=0 f y ( a , b ) = 0 ( ∇ f = 0 ) (\nabla f=\mathbf{0}) ( ∇ f = 0 )
Local max/min and critical point
If f f f ( a , b ) (a,b) ( a , b ) ( a , b ) (a,b) ( a , b )
Second derivative test ( a , b ) (a,b) ( a , b )
D ( a , b ) = f x x ( a , b ) f y y ( a , b ) − [ f x y ( a b ) ] 2 D(a,b) = f_{xx}(a,b)f_{yy}(a,b) - [f_{xy}(ab)]^2 \newline D ( a , b ) = f x x ( a , b ) f y y ( a , b ) − [ f x y ( a b ) ] 2 = ∣ f x x f x y f y x f y y ∣ = \begin{vmatrix} f_{xx} & f_{xy} \cr f_{yx} & f_{yy} \end{vmatrix} = ∣ ∣ ∣ ∣ ∣ f x x f y x f x y f y y ∣ ∣ ∣ ∣ ∣ D > 0 D>0 D > 0 f x x ( a , b ) > 0 f_{xx}(a,b)>0 f x x ( a , b ) > 0 f ( a , b ) f(a,b) f ( a , b ) D > 0 D>0 D > 0 f x x ( a , b ) < 0 f_{xx}(a,b)<0 f x x ( a , b ) < 0 f ( a , b ) f(a,b) f ( a , b ) D < 0 D<0 D < 0 f ( a , b ) f(a,b) f ( a , b )
Extreme value theorem for functions of two variables
If f f f D ∈ R 2 D\in \R^2 D ∈ R 2 f f f D D D
Closed boundary method
1. Find the values of f f f f f f D D D f f f D D D
Chain rule, directional derivative, and gradient vector are not covered in MATH 126 but covered in MATH 324 .
MATH 126 covers double integrals in Cartesian coordinates and polar coordinates with applications. They are reviewed in MATH 324 .
Linear and quadratic approximations
Description
Equations
First Taylor polynomial
T 1 ( x ) ≈ f ( b ) + f ′ ( b ) ( x − b ) T_1(x) \approx f(b) + f'(b)(x-b) T 1 ( x ) ≈ f ( b ) + f ′ ( b ) ( x − b )
Tangent line error
∣ E 1 ∣ = ∣ f ( x ) − [ f ( b ) + f ′ ( b ) ( x − b ) ] ∣ \lvert E_1 \rvert = \bigg\lvert f(x) - [f(b)+f'(b)(x-b)] \bigg\rvert ∣ E 1 ∣ = ∣ ∣ ∣ ∣ ∣ f ( x ) − [ f ( b ) + f ′ ( b ) ( x − b ) ] ∣ ∣ ∣ ∣ ∣
Tangent line error bound
∣ f ′ ′ ( t ) ∣ ≤ M ∣ E 1 ∣ ≤ M 2 ∣ x − b ∣ 2 \lvert f''(t) \rvert \le M \newline \lvert E_1 \rvert \le \dfrac{M}{2}\lvert x-b \rvert^2 ∣ f ′ ′ ( t ) ∣ ≤ M ∣ E 1 ∣ ≤ 2 M ∣ x − b ∣ 2
Second Taylor polynomial
T 2 ( x ) = f ( b ) + f ′ ( b ) ( x − b ) + 1 2 f ′ ′ ( b ) ( x − b ) 2 T_2(x) = f(b) + f'(b)(x-b)+\frac{1}{2}f''(b)(x-b)^2 T 2 ( x ) = f ( b ) + f ′ ( b ) ( x − b ) + 2 1 f ′ ′ ( b ) ( x − b ) 2
Quadratic approximation error
∣ E 2 ∣ = ∣ f ( x ) − T 2 ( x ) ∣ \lvert E_2 \rvert = \lvert f(x) - T_2(x) \rvert ∣ E 2 ∣ = ∣ f ( x ) − T 2 ( x ) ∣
Quadratic approximation error bound
∣ E 2 ∣ ≤ M 6 ∣ x − b ∣ 3 \lvert E_2 \rvert \le \dfrac{M}{6}\lvert x-b \rvert^3 ∣ E 2 ∣ ≤ 6 M ∣ x − b ∣ 3
Taylor polynomial and series
Description
Equations
n n n T n ( x ) = ∑ k = 0 n f ( k ) ( b ) k ! ( x − b ) k T_n(x) = \displaystyle\sum\limits_{k=0}^{n} \dfrac{f^{(k)}(b)}{k!} (x-b)^k \newline T n ( x ) = k = 0 ∑ n k ! f ( k ) ( b ) ( x − b ) k = f ( b ) + f ′ ( b ) ( x − a ) + f ′ ′ ( b ) 2 ! ( x − a ) 2 + … + f ( n ) ( b ) n ! ( x − b ) n = f(b) + f'(b)(x-a) + \dfrac{f''(b)}{2!}(x-a)^2 + … + \dfrac{f^{(n)}(b)}{n!} (x-b)^n = f ( b ) + f ′ ( b ) ( x − a ) + 2 ! f ′ ′ ( b ) ( x − a ) 2 + … + n ! f ( n ) ( b ) ( x − b ) n
Taylor inequality ( ∣ f ( n + 1 ) ( t ) ∣ ≤ M ) (\lvert f^{(n+1)}(t) \rvert \le M) ( ∣ f ( n + 1 ) ( t ) ∣ ≤ M )
∣ f ( x ) − T n ( x ) ∣ ≤ M ( n + 1 ) ! ∣ x − b ∣ n + 1 \lvert f(x) - T_n(x) \rvert \le \dfrac{M}{(n+1)!} \lvert x-b \rvert^{n+1} ∣ f ( x ) − T n ( x ) ∣ ≤ ( n + 1 ) ! M ∣ x − b ∣ n + 1
Taylor series
lim n → ∞ T n ( x ) \lim\limits_{n\to\infin}T_n(x) \newline n → ∞ lim T n ( x ) = lim n → ∞ ∑ k = 0 n f ( k ) ( b ) k ! ( x − b ) k = \displaystyle\lim\limits_{n\to\infin}\sum\limits_{k=0}^{n} \dfrac{f^{(k)}(b)}{k!} (x-b)^k \newline = n → ∞ lim k = 0 ∑ n k ! f ( k ) ( b ) ( x − b ) k = ∑ k = 0 ∞ f ( k ) ( b ) k ! ( x − b ) k = \sum\limits_{k=0}^{\infin} \dfrac{f^{(k)}(b)}{k!} (x-b)^k = k = 0 ∑ ∞ k ! f ( k ) ( b ) ( x − b ) k
Taylor series of exponential function
e x = ∑ k = 0 ∞ x k k ! e^x = \displaystyle\sum\limits_{k=0}^{\infin} \dfrac{x^{k}}{k!} e x = k = 0 ∑ ∞ k ! x k
Taylor series of sine
cos ( x ) = ∑ k = 0 ∞ ( − 1 ) k x 2 k ( 2 k ) ! \cos(x) = \displaystyle\sum\limits_{k=0}^{\infin} (-1)^k \dfrac{x^{2k}}{(2k)!} cos ( x ) = k = 0 ∑ ∞ ( − 1 ) k ( 2 k ) ! x 2 k
Taylor series of cosine
sin ( x ) = ∑ k = 0 ∞ ( − 1 ) k x 2 k + 1 ( 2 k + 1 ) ! \sin(x) = \displaystyle\sum\limits_{k=0}^{\infin} (-1)^k \dfrac{x^{2k+1}}{(2k+1)!} sin ( x ) = k = 0 ∑ ∞ ( − 1 ) k ( 2 k + 1 ) ! x 2 k + 1
Geometric series as Taylor series x ∈ ( − 1 , 1 ) x\in(-1,1) x ∈ ( − 1 , 1 )
1 1 − x = ∑ k = 0 ∞ x k \dfrac{1}{1-x} = \displaystyle\sum\limits_{k=0}^{\infin} x^k 1 − x 1 = k = 0 ∑ ∞ x k