벡터의 미분

Vector Function : r (t)

component function

r (t) =[f (t), g (t), h (t)] = f (t) i + g (t) j + h (t) k

Vector Field

F (x, y, z) = P (x, y, z) i + Q (x, y, z) j + R (x, y, z) k

Underscript[lim, t→t_0] r (t) =[Underscript[lim, t→t_0] f (t), Underscript[lim, t→t_0] g (t), Underscript[lim, t→t_0] h (t)]

r^′ (t) = Underscript[lim, Δt→0] (r (t + Δt) - r (t))/Δt

r^′ (t) = /t[f (t) i + g (t) j + h (t) k]

= f/ti +  g/tj +  h/tk

= f^′ (t) i + g^′ (t) j + h^′ (t) k

(c u)^′ = c u^′

(u + v)^′ = u^′ + v^′

(u·v)^′ = u^′·v + u·v^′

(u×v)^′ = u^′×v + u×v^′

/t[u (t) ·v (t)]

= /t[u_1 (t) v_1 (t) + u_2 (t) v_2 (t) + u_3 (t) v_3 (t)]

=[u_1^′ (t) v_1 (t) + u_2^′ (t) v_2 (t) + u_3^′ (t) v_3 (t)] +[u_1 (t) v_1^′ (t) + u_2 (t) v_2^′ (t) + u_3 (t) v_3^′ (t)]

= u^′ (t) ·v (t) + u (t) ·v^′ (t)

/t[u (t) ×v (t)]

= Underscript[lim, Δt→0] ((u + Δu) × (v + Δv) - u×v)/Δt

= Underscript[lim, Δt→0] (u×Δv + Δu×v + Δu×Δv)/Δt

= Underscript[lim, Δt→0] (u×Δv/Δt + Δu/Δt×v + Δu/Δt×Δv)

= u×v/t + u/t×v

/t (u×v)

= u×v/t + u/t×v

∂a/∂x = Underscript[lim, Δt→0] (a (x + Δx, y, z) - a (x, y, z))/Δx

∂a/∂y = Underscript[lim, Δt→0] (a (x, y + Δy, z) - a (x, y, z))/Δy

∂a/∂z = Underscript[lim, Δt→0] (a (x, y, z + Δz) - a (x, y, z))/Δz

A/t = (a_ij/t)

x^′ = (x_1, x_2, x_3)

A = ({{a_11, a_12, a_13}, {a_21, a_22, a_23}, {a_31, a_32, a_33}})

x^′Ax = (x_1, x_2, x_3) ({{a_11, a_12, a_13}, {a_21, a_22, a_23}, {a_31, a_32, a_33}}) ({{x_1}, {x_2}, {x_3}})

∂/∂x (x^′Ax) = 2Ax

∫At = ∫ (a_ij) t

∫_a^bAt = ∫_a^b (a_ij) t

공간 곡선의 접선벡터와 법선벡터

호의 길이

r (t) = x (t) i + y (t) j + z (t) k

s = ∫_a^b | Overscript[r, .] (t) | t = ∫_a^b (Overscript[x, .] (t)^2 + Overscript[y, .] (t)^2 + Overscript[z, .] (t)^2)^(1/2) t

s (t) = ∫_a^t | Overscript[r, .] (t) | t

s_n = Underoverscript[∑, i = 1, arg3] Overscript[P_ (i - 1) P_i, _] = Underoverscript[∑, i = 1, arg3][(Δx_i)^2 + (Δy_i)^2 + (Δz_i)^2]^1/2

= Underoverscript[∑, i = 1, arg3][(Δx_i/Δt)^2 + (Δy_i/Δt)^2 + (Δz_i/Δt)^2]^1/2Δt→s

s = ∫_a^b | r/t | t = ∫_a^b ((x)^2 + (y)^2 + (z)^2)^(1/2)

s (t) = ∫_a^t (Overscript[x, .]^2 + Overscript[y, .]^2 + Overscript[z, .]^2)^(1/2) t

접선벡터 (tangent vector)

Overscript[r, .] (t) = Underscript[lim, Δt→0] (r (t + Δt) - r (t))/Δt

T (t) = (Overscript[r, .] (t))/(| Overscript[r, .] (t) |)

법선벡터 (normal vector)

T (t) ·Overscript[T, .] (t) = 0

N (t) = (Overscript[T, .] (t))/(| Overscript[T, .] (t) |)

접선과 법평면

P (x (t_0), y (t_0), z (t_0))

Q (x (t_0 + Δt), y (t_0 + Δt), z (t_0 + Δt))

(x (t) - x (t_0))/(x (t_0 + Δt) - x (t_0)) = (y (t) - y (t_0))/(y (t_0 + Δt) - y (t_0)) = (z (t) - z (t_0))/(z (t_0 + Δt) - z (t_0))

(x (t) - x (t_0))/(x (t_0 + Δt) - x (t_0))/Δt = (y (t) - y (t_0))/(y (t_0 + Δt) - y (t_0))/Δt = (z (t) - z (t_0))/(z (t_0 + Δt) - z (t_0))/Δt

tangent vector

(x (t) - x (t_0))/(Overscript[x, .] (t_0)) = (y (t) - y (t_0))/(Overscript[y, .] (t_0)) = (z (t) - z (t_0))/(Overscript[z, .] (t_0))

normal plane

Overscript[x, .] (t_0) (x (t) - x (t_0)) + Overscript[y, .] (t_0) (y (t) - y (t_0)) + Overscript[z, .] (t_0) (z (t) - z (t_0)) = 0

Tangent Vector

T = r/s = r/t/s/t = Overscript[r, .]/(| Overscript[r, .] |)

Normal Vector

N = T^′/(| T^′ |)

| T^′ | = θ/s = Underscript[lim, Δt→0] Δθ/Δs

| T^′ | 는 곡선의 곡률 k

k = | T/s | = | /s (r/s) | = | ^2r/s^2 |

속도와 가속도

Overscript[v, _] = (s_2 - s_1)/(t_2 - t_1) = Δs/Δt

v = Underscript[lim, Δt→0] Δs/Δt

Overscript[a, _] = (v_2 - v_1)/(t_2 - t_1) = Δv/Δt

a = Underscript[lim, Δt→0] Δv/Δt

string

r (t) = x (t) i + y (t) j + z (t) k

s = ∫_a^b | Overscript[r, .] (t) | t = ∫_a^b ((x/t)^2 + (y/t)^2 + (z/t)^2)^(1/2) t

velocity

v (t) = Overscript[r, .] (t) = x/ti + y/tj + z/tk

acceleration

a (t) = Overscript[v, .] (t) = ^2x/t^2i + ^2y/t^2j + ^2z/t^2k