Unit 5 Lecture Guide .pdf - MTH 202u2013Calculus III Unit...

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MTH 202–Calculus III Unit 5: Vector Calculus Contents 1 Vector Fields 2 1.1 Formal Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Conservative Vector Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 Divergence and Curl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2 Line Integrals 3 2.1 Physical Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2 Evaluating a Line Integral WRT Arc Length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.3 Another Physical Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.4 Line Integrals WRT x , y , or z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.5 Piecewise Smooth Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 3 Work Integrals 6 3.1 Line Integrals of a VVF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 3.2 Work Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 4 Fundamental Theorem of Line Integrals 9 5 Green’s Theorem 11 5.1 Simply Connected Regions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 5.2 Area of a Region Using Green’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 5.3 Multiply Connected Regions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 6 Surface Integrals 13 6.1 Physical Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 6.2 Formal Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 6.3 Surface Area Redux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 7 Flux Integrals 15 7.1 Orientable Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 7.2 Physical Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 7.3 Evaluating Flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 8 Divergence Theorem 18 8.1 Sources and Sinks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 8.2 Informal Viewpoint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 8.3 Formal Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 9 Stokes’ Theorem 19
Harper College Dept. of Mathematical Sciences Unit 5: Vector Calculus Section 1 1 Vector Fields Functions come in all sizes and shapes so to speak. Many of the functions we deal with take real values as their inputs and produce real values as their outputs. But we have seen this semester functions that can take multiple inputs (multivariate functions) to produce a real value and also functions that can take real values and produce a vector. In this section, we learn about a new type of function that take points in 2-space or 3-space and produce vectors. 1.1 Formal Definition DEF’N: A vector field is a function that assigns each point P in a region of 2-space or 3-space a vector F ( P ). Notice that F is a vector-valued function. Its scalar components are determined by scalar functions of ( x, y ) or ( x, y, z ). F ( x, y ) = f ( x, y ) i + g ( x, y ) j F ( x, y, z ) = f ( x, y, z ) i + g ( x, y, z ) j + h ( x, y, z ) k RECALL: If Φ is a function of several variables, then Φ = Φ ∂x i + Φ ∂y j Φ = Φ ∂x i + Φ ∂y j + Φ ∂z k This shows that the gradient is actually a vector field. We call this the gradient field of Φ. Gradient Field Sketch the gradient field of Φ( x, y ) = x 2 + y . 1.2 Conservative Vector Fields DEF’N: A vector field F is said to be conservative in a region provided it is the gradient field for some function Φ in that region. In other words, Φ = F ( x, y ) (2-space) Φ = F ( x, y, z ) (3-space) The function Φ is called a potential function for F in that region. Conservative Values Determine if the given vector field is conservative. F ( x, y ) = 2 xy i + x 2 j Page 2 of 20
Harper College Dept. of Mathematical Sciences Unit 5: Vector Calculus Section 2

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