Calc 3 Flashcards
Tangent Line
A line that touches a curve at only 1 point and determines the instantaneous rate of change
A line that touches a curve at only 1 point and determines the instantaneous rate of change
Tangent Line
r(t) = c(r’(t))
two vectors are parallel
two vectors are parallel
r(t) = c(r’(t))
Unit Tangent Formula
T(t) = r’(t) / ||r’(t)|| (tangent vector divided by magnitude of tangent vector)
T(t) = r’(t) / ||r’(t)|| (tangent vector divided by magnitude of tangent vector)
Unit Tangent Formula
Principle Normal Vector
N(t) = T’(t) / ||T’(t)|| ( derivative of unit tangent divided by derivative unit tangent magnitude)
N(t) = T’(t) / ||T’(t)|| ( derivative of unit tangent divided by derivative unit tangent magnitude)
Principle Normal Vector
Equation for osculating plane
N(t) x T(t) = cross product of Principle Normal Vector and Unit Tangent
N(t) x T(t) = cross product of Principle Normal Vector and Unit Tangent
Equation for osculating plane
Arc Length Formula
Intergral of : square root ( x’(t) + y’(t) + z’(t) )
Intergral of : square root ( x’(t) + y’(t) + z’(t) )
Arc Length Formula
||v(t)|| = ||r’(t)||
speed at time t
speed at time t
||v(t)|| = ||r’(t)||
Curvature of a plane curve
k = |y’’| / ( 1 + (y’)^2 ) ^ 3/2
k = |y’’| / ( 1 + (y’)^2 ) ^ 3/2
Curvature of a plane curve
r(t) = x(t) + y(t) = given vector function curvature is
k = |x’y’’ - y’x’’| / ( (x’)^2 + (y’)^2 )^ 3/2
k = |x’y’’ - y’x’’| / ( (x’)^2 + (y’)^2 )^ 3/2
r(t) = x(t) + y(t) = vector function curvature
Curvature of a space curve
k = ||dT/dt|| / (ds/dt)
k = ||dT/dt|| / (ds/dt)
Curvature of a space curve
(ds/dt)
velocity
velocity notation derivative
(ds/dt)
aT = Tangential Acceleration
T dot a(t) = depends only on the change of speed
T dot a(t) = depends only on the change of speed
aT = Tangential Acceleration
aN = Normal component acceleration
|| T x a(t) || = depends on speed and curvature
|| T x a(t) || = depends on speed and curvatur
aN = Normal component acceleration
curvature of the path of acceleration
k = ||v(t) x a(t)|| / (ds/dt)^3 = cross product magnitude of velocity and acceleration divided by velocity ^ 3
k = ||v(t) x a(t)|| / (ds/dt)^3 = cross product magnitude of velocity and acceleration divided by velocity ^ 3
curvature of the path of acceleration
Directional Derivative
Gives the rate of change of F in the direction U.
(F) Gradient Vector * (U) unit Vector of Direction
Gives the rate of change of F in the direction U.
(F) Gradient Vector * (U) unit Vector of Direction
Directional Derivative
Gradient Vector
1st Partial Derivative in respect to (x, y, z)
1st Partial Derivative in respect to (x, y, z) forms what vector?
Gradient Vector
|| vT(x, y) || = Magnitude of the Directional Derivative implies
Rate of change at (x, y) (rate of increase at (x, y))
Rate of change at (x, y) (rate of increase at (x, y))
|| vT(x, y) || = Magnitude of the Directional Derivative
- || vT(x, y) || = Magnitude of the Directional Derivative negative implies
Rate of change (x, y) (rate of decrease at (x, y))
Rate of change (x, y) (rate of decrease at (x, y))
- || vT(x, y) || = Magnitude of the Directional Derivative negative
