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Chapter 9: Vector Differential Calculus 1 9.1. Vector Functions of One Variable -- a vector, each component of which is a function of the same variable i.e., F(t) = x(t) i + y(t) j + z(t) k, where x(t), y(t), z(t): component functions t : a variable e.g., ◎ Definition 9.1: Vector function of one variable

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2 。 F(t) is continuous at some t 0 if x(t), y(t), z(t) are all continuous at t 0 。 F(t) is differentiable if x(t), y(t), z(t) are all differentiable ○ Derivative of F(t): e.g.,

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3 ○ Curve: C(x(t), y(t), z(t)), in which x(t), y(t), z(t): coordinate functions x = x(t), y = y(t), z = z(t): parametric equations F(t)= x(t)i + y(t)j + z(t)k: position vector pivoting at the origin Tangent vector to C: Length of C:

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4 ○ Example 9.2: Position vector: Tangent vector: Length of C:

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5 ○ Distance function: t(s): inverse function of s(t) ○ Let Unit tangent vector:

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6 。 Example 9.3: Position function: Inverse function:

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7 Unit tangent vector:

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8 ○ Assuming that the derivatives exist, then (1) (2) (3) (4) (5)

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9 9.2. Velocity, Acceleration, Curvature, Torsion A particle moving along a path has position vector Distance function: ◎ Definition 9.2: Velocity: (a vector) tangent to the curve of motion of the particle Speed : (a scalar) the rate of change of distance w.r.t. time

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10 Acceleration: or (a vector) the rate of change of velocity w.r.t. time ○ Example 9.4: The path of the particle is the curve whose parametric equations are

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11 Velocity: Speed: Acceleration: Unit tangent vector:

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12 ○ Definition 9.4: Curvature (a magnitude): the rate of change of the unit tangent vector w.r.t. arc length s For variable t,

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13 ○ Example 9.7: Curve C: t > 0 Position vector:

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14 Tangent vector: Unit tangent vector: Curvature:

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15 ◎ Definition 9.5: Unit Normal Vector i) ii) Differentiation

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16 ○ Example 9.8: Position vector: t > 0 Write as a function of arc length s (Example 9.7) Solve for t, Position vector:

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17 Unit tangent vector: Curvature:

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18 Unit normal vector:

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19 9.2.1 Tangential and Normal Components of Acceleration

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20 ◎ Theorem 9.1: where Proof:

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21 ○ Example 9.9: Compute and for curve C with position vector Velocity: Speed: Tangential component: Acceleration vector:

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22 Normal component: Acceleration vector: Since, curvature: Unit tangent vector: Unit normal vector:

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23 ◎ Theorem 9.2: Curvature Proof:

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24 ○ Example 9.10: Position function:

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25 9.2.3 Frenet Formulas Let Binormal vector: T, N, B form a right-handed rectangular coordinate system This system twists and changes orientation along curve

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26 ○ Frenet formulas : The derivatives are all with respect to s. (i) From Def. 9.5, (ii) is inversely parallel to N Let : Torsion

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27 (iii) (a) (b) (c) ＊ Torsion measures how (T, N, B) twists along the curve

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28 12.3 Vector Fields and Streamlines ○ Definition 9.6: Vector Field -- (3-D) A vector whose components are functions of three variables -- (2-D) A vector whose components are functions of two variables

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29 。 A vector filed is continuous if each of its component functions is continuous. 。 A partial derivative of a vector field -- the vector fields obtained by taking the partial derivative of each component function e.g.,

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30 ◎ Definition 9.7: Streamlines F: vector field defined in some 3-D region Ω : a set of curves with the property that through each point P of Ω, there passes exactly one curve from The curves in are streamlines of F if at each point in Ω, F curve in passing through is tangent to the

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31 ○ Vector filed: : Streamline of F Parametric equations -- Position vector -- Tangent vector at --

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32 is also tangent to C at //

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33 ○ Example 9.11: Find streamlines Vector field: From Integrate Solve for x and y Parametric equations of the streamlines

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34 Find the streamline through (-1, 6, 2).

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35 9.4. Gradient Field and Directional Derivatives ◎ Definition 9.8: Scalar field: a real-valued function e.g. temperature, moisture, pressure, hight Gradient of : a vector field

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36 e.g., 。 Properties: ○ Definition 9.9: Directional derivative of in the direction of unit vector

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37 ◎ Theorem 9.3: Proof: By the chain rule

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38 ○ Example 9.13:

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39 ◎ Theorem 9.4: has its 1. Maximum rate of change,, in the direction of 2. Minimum rate of change,, in the direction of Proof: Max.: Min.:

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40 ○ Example 9.4: The maximum rate of change at The minimum rate of change at

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41 9.4.1. Level Surfaces, Tangent Planes, and Normal Lines ○ Level surface of : a locus of points e.g., Sphere (k > 0) of radius Point (k = 0), Empty (k < 0)

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42 ○ Tangent Plane at point to Normal vector: the vector perpendicular to the tangent plane

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43 ○ Theorem 9.5: Gradient normal to at point on the level surface Proof: Let : a curve passing point P on surface C lies on

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44 normal to This is true for any curve passing P on the surface. Therefore, normal to the surface

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45 ○ Find the tangent plane to Let (x, y, z): any point on the tangent plane orthogonal to the normal vector The equation of the tangent plane:

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46 ○ Example 9.16: Consider surface Let The surface is the level surface Gradient vector: Tangent plane at

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48 9.5. Divergence and Curl ○ Definition 9.10: Divergence (scalar field) e.g.,

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49 ○ Definition 9.11: Curl (vector field) e.g.,

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50 ○ Del operator: 。 Gradient: 。 Divergence: 。 Curl:

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51 ○ Theorem 9.6: Proof:

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52 ◎ Theorem 9.7: Proof:

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FORMULA ○ Position vector of curve F(t)= x(t)i + y(t)j + z(t)k 。 Distance function: 。 Unite tangent vector: where C(x(t), y(t), z(t))

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○ Velocity: Speed : Acceleration: or, where=, = ○ Curvature:

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○ Unit Normal Vector: ○ Binormal vector: Torsion ○ Frenet formulas: ○ Vector filed: Streamline:

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○ Scalar field: Gradient: ○ Directional derivative: ○ Divergence: ○ Curl: ○

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