Chapter 9 Vector Differential Calculus 9 1 Vector

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Chapter 9: Vector Differential Calculus 9. 1. Vector Functions of One Variable ◎ Definition

Chapter 9: Vector Differential Calculus 9. 1. Vector Functions of One Variable ◎ Definition 9. 1: Vector function 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. , 1

。F(t) is continuous at some t 0 if x(t), y(t), z(t) are all continuous

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

○ Curve: C(x(t), y(t), z(t)), in which x(t), y(t), z(t): coordinate functions x =

○ 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: 3

○ Example 9. 2: Position vector: Tangent vector: Length of C: 4

○ Example 9. 2: Position vector: Tangent vector: Length of C: 4

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

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

。 Example 9. 3: Position function: Inverse function: 6

。 Example 9. 3: Position function: Inverse function: 6

Unit tangent vector: 7

Unit tangent vector: 7

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

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

9. 2. Velocity, Acceleration, Curvature, Torsion A particle moving along a path has position

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 9

Acceleration: or (a vector) the rate of change of velocity w. r. t. time

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 10

Velocity: Speed: Acceleration: Unit tangent vector: 11

Velocity: Speed: Acceleration: Unit tangent vector: 11

○ Definition 9. 4: Curvature (a magnitude): the rate of change of the unit

○ Definition 9. 4: Curvature (a magnitude): the rate of change of the unit tangent vector w. r. t. arc length s For variable t, 12

○ Example 9. 7: Curve C: t>0 Position vector: 13

○ Example 9. 7: Curve C: t>0 Position vector: 13

Tangent vector: Unit tangent vector: Curvature: 14

Tangent vector: Unit tangent vector: Curvature: 14

◎ Definition 9. 5: Unit Normal Vector i) ii) Differentiation 15

◎ Definition 9. 5: Unit Normal Vector i) ii) Differentiation 15

○ Example 9. 8: Position vector: t > 0 Write as a function of

○ Example 9. 8: Position vector: t > 0 Write as a function of arc length s (Example 9. 7) Solve for t, Position vector: 16

Unit tangent vector: Curvature: 17

Unit tangent vector: Curvature: 17

Unit normal vector: 18

Unit normal vector: 18

9. 2. 1 Tangential and Normal Components of Acceleration 19

9. 2. 1 Tangential and Normal Components of Acceleration 19

◎ Theorem 9. 1: where Proof: 20

◎ Theorem 9. 1: where Proof: 20

○ Example 9. 9: Compute and for curve C with position vector Velocity: Speed:

○ Example 9. 9: Compute and for curve C with position vector Velocity: Speed: Tangential component: Acceleration vector: 21

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

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

◎ Theorem 9. 2: Curvature Proof: 23

◎ Theorem 9. 2: Curvature Proof: 23

○ Example 9. 10: Position function: 24

○ Example 9. 10: Position function: 24

9. 2. 3 Frenet Formulas Let Binormal vector: T, N, B form a right-handed

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 25

○ Frenet formulas: The derivatives are all with respect to s. (i) From Def.

○ Frenet formulas: The derivatives are all with respect to s. (i) From Def. 9. 5, (ii) is inversely parallel to N Let : Torsion 26

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

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

12. 3 Vector Fields and Streamlines ○ Definition 9. 6: Vector Field -- (2

12. 3 Vector Fields and Streamlines ○ Definition 9. 6: Vector Field -- (2 -D) A vector whose components are functions of two variables -- (3 -D) A vector whose components are functions of three variables 28

。 A vector filed is continuous if each of its component functions is continuous.

。 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. , 29

◎ Definition 9. 7: Streamlines F: vector field defined in some 3 -D region

◎ 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 is tangent to the curve in passing through 30

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

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

is also tangent to C at // 32

is also tangent to C at // 32

○ Example 9. 11: Find streamlines Vector field: From Integrate Solve for x and

○ Example 9. 11: Find streamlines Vector field: From Integrate Solve for x and y Parametric equations of the streamlines 33

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

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

9. 4. Gradient Field and Directional Derivatives ◎ Definition 9. 8: Scalar field: a

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 35

e. g. , 。 Properties: ○ Definition 9. 9: Directional derivative of in the

e. g. , 。 Properties: ○ Definition 9. 9: Directional derivative of in the direction of unit vector 36

◎ Theorem 9. 3: Proof: By the chain rule 37

◎ Theorem 9. 3: Proof: By the chain rule 37

○ Example 9. 13: 38

○ Example 9. 13: 38

◎ Theorem 9. 4: has its 1. Maximum rate of change, , in the

◎ 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. : 39

○ Example 9. 4: The maximum rate of change at The minimum rate of

○ Example 9. 4: The maximum rate of change at The minimum rate of change at 40

9. 4. 1. Level Surfaces, Tangent Planes, and Normal Lines ○ Level surface of

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) 41

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

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

○ Theorem 9. 5: Gradient normal to at point on the level surface Proof:

○ Theorem 9. 5: Gradient normal to at point on the level surface Proof: Let : a curve passing point P on surface C lies on 43

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

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

○ Find the tangent plane to Let (x, y, z): any point on the

○ 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: 45

○ Example 9. 16: Consider surface Let The surface is the level surface Gradient

○ Example 9. 16: Consider surface Let The surface is the level surface Gradient vector: Tangent plane at 46

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9. 5. Divergence and Curl ○ Definition 9. 10: Divergence (scalar field) e. g.

9. 5. Divergence and Curl ○ Definition 9. 10: Divergence (scalar field) e. g. , 48

○ Definition 9. 11: Curl (vector field) e. g. , 49

○ Definition 9. 11: Curl (vector field) e. g. , 49

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

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

○ Theorem 9. 6: Proof: 51

○ Theorem 9. 6: Proof: 51

◎ Theorem 9. 7: Proof: 52

◎ Theorem 9. 7: Proof: 52

FORMULA ○ Position vector of curve C(x(t), y(t), z(t)) F(t)= x(t)i + y(t)j +

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

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

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

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

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

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

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