Topic 2-2: Derivatives of Vector Functions. Textbook: Section 13.2, 13.4

Topic 2-2: Derivatives of Vector Functions Textbook: Section 13.2, 13.4

Warm-Up: Parametrization of Circles Each of the following vector functions describe the position of an object traveling around the same circle in different ways: r 1 (t) = 3 cos t î + 3 sin t ĵ r 2 (t) = 3 cos(2t) î + 3 sin(2t) ĵ r 3 (t) = 3 cos(t + π/2) î + 3 sin(t + π/2) ĵ

Warm-Up: Parametrization of Circles What is the radius of the circle? Enter an integer between 1 and 10. Enter 100 if you don t know how to answer the question without guessing.

Warm-Up: Parametrization of Circles Are all objects in the same position at t = 0? 1. Yes. 2. No. 3. I don t know how to answer this question.

Warm-Up: Parametrization of Circles Are all objects traveling in the same direction? 1. Yes. 2. No. 3. I don t know how to answer this question.

Warm-Up: Parametrization of Circles Which is the parameterization of the fastest-traveling object? 1. r 1 (t) 2. r 2 (t) 3. r 3 (t) 4. They are all traveling at the same speed. 5. I don t know how to answer this question.

Warm-up: Parameterizing Graphs of Functions Which of the following vector functions does not parametrize the cubic curve described by y = x 3? 1. r(t) = t î + t 3 ĵ 2. r(u) = u î + u 3 ĵ 3. r(t) = t 3 î + t ĵ 4. r(t) = t 1/3 î + t ĵ 5. r(x) = x î + x 3 ĵ 6. I don t know how to answer this question.

Big Ideas Derivatives (and integrals) of vector functions are computed by differentiating (or integrating) each coordinate function. If a vector function r(t) represents the position of an object at time t, then r (t) = v(t) represents velocity, r (t) = v(t) = v(t) represents speed, and r (t) = v (t) = a(t) represents acceleration. The derivative r (t) is tangent to the curve parameterized by the vector function r(t). Other important tangent vectors: the unit tangent vector ˆT, and the vector differential dr.

Continuity, Differentiation, and Integration of Vector Functions The behavior of a vector function r(t) = x(t), y(t), z(t) depends only on the coordinate functions. If x(t), y(t), and z(t) are continuous, then so is r(t). If x(t), y(t), and z(t) are differentiable, then so is r(t), and: r (t) = x (t), y (t), z (t) If x(t), y(t), and z(t) are integrable, then so is r(t), and: ˆ ˆ ˆ ˆ r(t) dt = x(t) dt, y(t) dt, z(t) dt

Recall: the Derivative from Calc I The derivative of a scalar-valued function of a single variable: Is defined as the limit of a difference quotient: f (t) = lim h 0 f (t + h) f (t) h

Recall: the Derivative from Calc I The derivative of a scalar-valued function of a single variable: Is defined as the limit of a difference quotient: f (t) = lim h 0 f (t + h) f (t) h Geometrically, gives the slope of the tangent line to the graph of the function. Represents an instantaneous rate of change. If the function represents position, the derivative represents velocity, and the second derivative represents acceleration.

Definition of the Vector Derivative The derivative of a vector function is defined as the limit: r r (t) = lim h 0 h The vector r = r(t + h) r(t) is the displacement vector from the position r(t) to the position r(t + h). The displacement vector is multiplied by the scalar 1/h, where h is the increment (t + h) t in the parameter domain.

Geometry of the Vector Derivative If the derivative r (t) exists: r (t) points in the direction along the curve in which t increases. If r (t) 0, then r (t) spans a tangent line to the curve described by the vector function r(t). For this reason, r (t) is often called the tangent vector. r (t) represents the instantaneous rate of change of the position vector r(t) this change can occur in both the magnitude and the direction of r(t).

Position, Velocity, Acceleration If r(t) represents position at time t, then: r (t) = dr is the instantaneous rate of change of dt position with respect to time r (t) = v(t) is velocity. The magnitude of velocity is speed v(t) = v(t). r (t) = dr is the instantaneous rate of change of dt velocity with respect to time r (t) = a(t) is acceleration.

Discussion Question: Direction of Acceleration For the given curve: sketch velocity vectors v(t) = r (t) at the indicated points (magnitude doesn t matter, just direction). Between the terminal points of every two adjacent velocity vectors, sketch the displacement vector v. Imagine the points getting closer and closer together. What direction will v point, relative to the curve?

Vector Derivatives and Tangent Lines Calc I: f (t 0 ) is the slope of the tangent line to the graph of the function y = f (t) at the point ( t 0, f (t 0 ) ) Calc III: If r (t 0 ) 0, it is a direction vector for the curve parametrized by r(t), at the point ( x(t 0 ), y(t 0 ), z(t 0 ) ). Vector equation of the tangent line to r(t) at time t 0 : R(s) = r(t 0 ) + sr (t 0 )

Differentiation Rules for Vector Functions... Constant Rule: d dt C = 0 Linearity: d [ ] ar(t) + bs(t) = ar (t) + bs (t) dt Chain Rule: d ( ) dt r f (t) = f (t)r (f (t)) continued...

... Three Product Rules! Multiplication by Scalar Function: Dot Product: d dt f (t)r(t) = f (t)r(t) + f (t)r (t) d [ ] [ ] [ ] r(t) s (t) = r (t) s (t) + r(t) s (t) dt Cross Product : d [ ] [ ] [ ] r(t) s (t) = r (t) s (t) + r(t) s (t) dt

If r(t) Is Constant, then r(t) and r (t) are Orthogonal Suppose r(t) = c. Using the fact that r(t) 2 = r(t) r(t), and the product rule for the dot product of vector functions: d [ ] r(t) r(t) = d dt dt c2 r (t) r(t) + r(t) r (t) = 0 [ ] 2 r (t) r(t) = 0 r (t) r(t) = 0 This is why position and velocity are orthogonal when traveling on a circle, and why velocity and acceleration are orthogonal when traveling along a circle at constant speed.

Unit Tangent Vector ˆT (t) If r (t) 0, then: ˆT (t) = r (t) r (t) ˆT (t) is called the unit tangent vector because: ˆT (t) is tangent to the curve. ˆT (t) = 1.

Example: Computing ˆT (t) Compute ˆT (t) for the parabola y = x 2, parameterized as: r(t) = t î + t 2 ĵ.

There s a third vector, the binormal vector ˆB(t) = ˆT (t) ˆN(t). Together, ˆT, ˆN, ˆB form a moving frame that can be used to express vectors in a natural way with respect to motion along a curve. Unit Normal Vector ˆN If ˆT (t) 0, then: ˆN(t) = ˆT (t) ˆT (t) ˆN(t) is called the unit normal vector because: ˆN(t) is orthogonal to ˆT (t), so is normal to the curve. ˆN(t) = 1. ˆN(t) is usually not fun to compute, although there are several formulas one can use in special cases.

Another Useful Tangent Vector: the Vector Differential d r (will be used in chapter 16) The vector differential (or vector line element) dr is a vector representing an infinitesimal change in position from a point (x, y, z) to the (very) nearby point (x + dx, y + dy, z + dz). So: The magnitude of dr is: dr = dx, dy, dz ds = dr = dx 2 + dy 2 + dz 2

Another Useful Tangent Vector: the Vector Differential d r (will be used in chapter 16) The vector differential (or vector line element) dr is a vector representing an infinitesimal change in position from a point (x, y, z) to the (very) nearby point (x + dx, y + dy, z + dz). So: The magnitude of dr is: dr = dx, dy, dz ds = dr = dx 2 + dy 2 + dz 2 The magnitude of dr is an infinitesimal version of the Pythagorean theorem. It gives a way of measuring distance along a curve.

Computing d r r(t) = x(t) î + y(t) ĵ + z(t) ˆk. dr = d(x (t)) î + d(y (t)) ĵ + d(z (t)) ˆk = x (t) dt î + y (t) dt ĵ + z (t) dt ˆk ( = x (t) î + y (t) ĵ + z (t) ˆk ) dt = r (t) dt = velocity multiplied by the infinitesimal time increment dt Recall from u-sub in Calc I/II: If u = f (t), then du = f (t)dt.

Clicker Question: Speed Review Are you ready to go? 1. Yes.

Clicker Question: Speed Review Which one of these words has a meaning which is very different than the others? 1. Parallel. 2. Orthogonal. 3. Normal. 4. Perpendicular. 5. I don t remember.

Clicker Question: Speed Review Which of the following is a vector? 1. v w 2. v w 3. v w 4. All of the above are vectors. 5. I don t remember.

Clicker Question: Speed Review To compute a vector projection, use the: 1. Dot product. 2. Cross product. 3. I don t remember.

Clicker Question: Speed Review To compute the area of a parallelogram with sides v and w, use: 1. Dot product: v w. 2. Cross product: v w. 3. Magnitude of cross product: v w. 4. I don t remember.

Clicker Question: Speed Review To determine whether v and w are orthogonal, use the: 1. Dot product: v w. 2. Cross product: v w. 3. Magnitude of cross product: v w. 4. I don t recall.

Clicker Question: Speed Review To generate a normal vector to a plane containing v and w, use the: 1. Dot product: v w. 2. Cross product: v w. 3. Magnitude of cross product: v w. 4. I ve got no memory of anything at all.