Topic 2.3: The Geometry of Derivatives of Vector Functions

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1 BSU Math 275 Notes Topic 2.3: The Geometry of Derivatives of Vector Functions Textbook Sections: 13.2 From the Toolbox (what you nee from previous classes): Be able to compute erivatives scalar-value functions of a single variable, an know that these erivatives represent slopes of tangent lines an instantaneous rates of change. Also, know how these erivatives are relate to velocity an acceleration (Calc I). Vector operations (vector aition, scalar multiplication, the ot an cross proucts), an compute the magnitue of a vector. Be able to evaluate a vector function at specific values of its parameter. Be able to compute erivatives of vector functions. Given a vector function representing position along a curve at time t, compute an answer questions about position, velocity, spee, an acceleration. Learning Objectives (New Skills) & Important Concepts Learning Objectives (New Skills): Compute the unit tangent vector ˆT. Fin equations of tangent lines to curves parameterize by vector functions. Apply any previous knowlege about vectors to vector functions (vector aition, scalar multiplication, magnitue, ot an cross proucts, fining angles, projections, etc).

2 Important Concepts: If r (t) 0, it spans a line tangent to the curve parameterize by r(t). For this reason, r (f ) is often calle the tangent vector. The unit tangent vector ˆT = r (t)/ r (t) is the vector function that has the same irection as the vector erivative r (t), an constant length ˆT = 1 for all values of the parameter t. Because r(t) an its erivatives are vectors, one can perform any vector operation with them. The Big Picture The vector erivative r (t) is formally efine as a limit: r (t) = lim r(t) t This is similar to the limit efinition of a erivative f (t) from Calc I, an, as is the case for erivatives of functions of a single variable, the vector erivative represents the instantaneous rate of change of the vector function r(t) with respect to the parameter t. If r(t) represents position an the parameter t represents time, the first erivative or r(t) is velocity, an the secon erivative is acceleration. When the erivative r (t) 0, it spans a line tangent to the curve at the point whose position is given by r(t). For this reason, r (t) is sometimes calle the tangent vector. There are also two other useful vectors that have the same irection as r (t), but ifferent magnitues. One is the unit tangent vector ˆT = r (t)/ r (t). The secon is the vector ifferential (or line element) r = r (t) t. 2

3 More Details There are many similarities between the Calc I erivative of a function f (t) an the Calc III erivative of a vector function r(t). For example: Both f (t) an r(t) are functions of a single parameter (input is a scalar). The erivative of r(t) is compute by taking the erivatives of the coorinate functions, using the methos learne in Calc I. The erivative f (t) is the slope of the tangent line to the graph of the function f (t); the erivative r (t) is the irection vector of the tangent line to the curve r(t) (provie r (t) 0). If r(t) represents position an the parameter t represents time, the first erivative or r(t) is velocity, an the secon erivative is acceleration. This is analogous to the interpretations of the erivatives of a single-variable position function f (t). Both f (t) an r (t) are efine as the limit of ifference quotients. The vector erivative r (t) is formally efine as the limit (if it exists): Derivative of a single-variable function (Calc I): Derivative of a vector function: f f (t) = lim t r r (t) = lim t One ifference between the two types of erivatives: f (t) is a scalar value function (output is a scalar); r (t) is a vector value function (output is a vector). 3

4 The vector r is the isplacement vector from a starting position r(t) = x(t), y(t), z(t) to a nearby position r(t + t) = x(t + t), y(t + t), z(t + t). This change in position is really just a change in each of the coorinates: r = r(t + t) r(t) = x(t + t), y(t + t), z(t + t) x(t), y(t), z(t) = x, y, z So, if the limit exists, the vector erivative is: r r (t) = lim t x, y, z = lim = lim t x t, lim y t, lim = x (t), y (t), z (t) z t Which means, vector erivatives can be compute by computing the erivatives of the components, using the methos learne in Calc I. Geometry of the vector erivative: If the erivative r (t) exists, then: If r (t) 0, then r (t) points in the irection along the curve in which t increases. If r (t) 0, then r (t) spans a tangent line to the curve escribe by the vector funtion r(t). For this reason, r (t) is often calle the tangent vector. r (t) represents the instantaneous rate of change of the position vector r(t) this change can occur in both the magnitue an the irection of r(t). A smooth curve is a curve that can be escribe by a position function r (t) that is continuous an non-zero for all t. 4

5 If r (t 0 ) 0, then a vector equation of the tangent line to r(t) at t = t 0 can be foun by using r (t 0 ) as the irection vector, an aing the position r(t 0 ): R(s) = r(t 0 ) + sr (t 0 ) A unit tangent vector ˆT is a vector of length 1 that spans a tangent line to the curve. If r (t) 0, then ˆT (t) = r (t)/ r (t). There are two aitional unit vector associate with curves the unit normal vector ˆN, an the unit binormal vector ˆB. The three vectors ˆT, ˆN, ˆB form the moving frame, an are use to escribe vectors in a way that is natural with respect to motion along the curve. For example, acceleration can be ecompose into its tangential an normal components: a(t) = aˆt ˆT + a ˆN. ˆN Another useful tangent vector is the vector ifferential (or vector line element) r. This is a vector representing an infinitesimal change in position from a point (x, y, z) to the (very) nearby point (x + x, y + y, z + z). So: r = x, y, z The magnitue of r is: s = r = x 2 + y 2 + z 2 which is an infinitesimal version of the Pythagorean theorem. So the vector ifferential gives a way of measuring istance along a curve. To compute the vector ifferential of the vector function r(t) = x(t) î + y(t) ĵ + z(t) ˆk : r = (x (t)) î + (y (t)) ĵ + (z (t)) ˆk = x (t) t î + y (t) t ĵ + z (t) t ˆk ( ) = x (t) î + y (t) ĵ + z (t) ˆk t = r (t) t = velocity times the infinitesimal time increment t Recall from u-sub in Calc I/II: If u = f (t), then u = f (t)t. 5

6 Differentiation rules for vector functions: Constant Rule t C = 0 Linearity [ ] a r(t) + b s(t) = a r (t) + b s (t) t Prouct Rules There are three: Multiplication by Scalar-value function: [ ] f (t) r(t) = f (t) r(t) + f (t) r (t) t Dot prouct of two vector functions: [ ] [ ] [ ] r(t) s(t) = r (t) s(t) + r(t) s (t) t Cross prouct of two vector functions: [ ] [ ] [ ] r(t) s(t) = r (t) s(t) + r(t) s (t) t 6

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