9/6/2005 section_2_5_The_Calculus_of_Vector_Fields_empty.doc 1/9 2-5 The Calculus of Scalar and Vector Fields (pp.33-55) Fields are functions of coordinate variables (e.g., x, ρ, θ) Q: How can we integrate or differentiate vector fields ?? A: There are many ways, we will study: 1. 4. 2. 5. 3. 6. A. The Integration of Scalar and Vector Fields 1. The Line Integral Jim Stiles The Univ. of Kansas Dept. of EECS 9/6/2005 section_2_5_The_Calculus_of_Vector_Fields_empty.doc 2/9 ⋅ A ∫A(rdc ) C Q1: HO: Differential Displacement Vectors A1: HO: The Differential Displacement Vectors for Coordinate Systems Q2: A2: HO: The Line Ingtegral Q3: HO: The Contour C A3: HO: Line Integrals with Complex Contours Q4: Jim Stiles The Univ. of Kansas Dept. of EECS 9/6/2005 section_2_5_The_Calculus_of_Vector_Fields_empty.doc 3/9 HO: Steps for Analyzing Line Integrals A4: Example: The Line Integral 2. The Surface Integral Another important integration is the surface integral: A(rds)⋅ ∫∫ s S Q1: A1: HO: Differential Surface Vectors HO: The Differential Surface Vectors for Coordinate Systems Q2: A2: HO: The Surface Integral Jim Stiles The Univ. of Kansas Dept. of EECS 9/6/2005 section_2_5_The_Calculus_of_Vector_Fields_empty.doc 4/9 Q3: HO: The Surface S A3: HO: Integrals with Complex Surfaces Q4: HO: Steps for Analyzing Surface Integrals A4: Example: The Surface Integral 3. The Volume Integral The third important integration is the volume integral—it’s the easiest of the 3! g rdv ∫∫∫ ( ) V Q1: Jim Stiles The Univ. of Kansas Dept. of EECS 9/6/2005 section_2_5_The_Calculus_of_Vector_Fields_empty.doc 5/9 A1: HO: The Differential Volume Element HO: The Volume V Example: The Volume Integral B. The Differentiation of Vector Fields 1. The Gradient The Gradient of a scalar field g (r ) is expressed as: ∇g (r ) ∇g (rr) = A( ) Q: A: HO: The Gradient Jim Stiles The Univ. of Kansas Dept. of EECS 9/6/2005 section_2_5_The_Calculus_of_Vector_Fields_empty.doc 6/9 Q: A: HO: The Gradient Operator in Coordinate Systems Q: The gradient of every scalar field is a vector field—does this mean every vector field is the gradient of some scalar field? A: HO: The Conservative Field Example: Integrating the Conservative Field 2. Divergence The Divergence of a vector field A(r ) is denoted as: ∇ ⋅A(r ) ∇⋅A(rgr) = ( ) Jim Stiles The Univ. of Kansas Dept. of EECS 9/6/2005 section_2_5_The_Calculus_of_Vector_Fields_empty.doc 7/9 Q: A: HO: The Divergence of a Vector Field Q: A: HO: The Divergence Operator in Coordinate Systems HO: The Divergence Theorem 3. Curl The Curl of a vector field A(r ) is denoted as: ∇ × A(r ) ∇×AB(rr) = ( ) Jim Stiles The Univ. of Kansas Dept. of EECS 9/6/2005 section_2_5_The_Calculus_of_Vector_Fields_empty.doc 8/9 Q: A: HO: The Curl Q: A: HO: The Curl Operator in Coordinate Systems HO: Stoke’s Theorem HO: The Curl of a Conservative Vector Field 4. The Laplacian ∇2 g (r ) Jim Stiles The Univ. of Kansas Dept. of EECS 9/6/2005 section_2_5_The_Calculus_of_Vector_Fields_empty.doc 9/9 C. Helmholtz’s Theorems ∇⋅AA()rrand/or ∇× ( ) Q: A: HO: Helmholtz’s Theorems Jim Stiles The Univ. of Kansas Dept. of EECS 9/6/2005 Differential Line Vector Elements.doc 1/4 Differential Displacement Vectors The derivative of a position vector r , with respect to coordinate value A (where A ∈{xyz,,,,,,ρ φθ r }) is expressed as: dr d ˆˆˆ =++()xyzaaaxyz ddAA ˆ dx()aaˆˆdy()ay dz() =++xz dddAAA ⎛⎞dxˆˆˆ⎛⎞dy ⎛⎞ dz =++⎜⎟aaaxyz⎜⎟ ⎜⎟ ⎝⎠dddAAA⎝⎠ ⎝⎠ Q: Immediately tell me what this incomprehensible result means or I shall be forced to pummel you ! A: The vector above describes the change in position vector r due to a change in coordinate variable A. This change in position vector is itself a vector, with both a magnitude and direction. Jim Stiles The Univ. of Kansas Dept. of EECS 9/6/2005 Differential Line Vector Elements.doc 2/4 For example, if a point moves such that its coordinate A changes from A to AA+∆ , then the position vector that describes that point changes from r to r + ∆A. ∆A r r + ∆A In other words, this small vector ∆A is simply a directed distance between the point at coordinate A and its new location at coordinate AA+∆ ! This directed distance ∆A is related to the position vector derivative as: dr ∆=∆AA d A ⎛⎞dxˆˆˆ⎛⎞dy ⎛⎞ dz =∆AAA⎜⎟aaaxyz +∆⎜⎟ +∆ ⎜⎟ ⎝⎠dddAAA⎝⎠ ⎝⎠ As an example, consider the case when A = ρ . Since x = ρ cosφ and y = ρ sinφ we find that: Jim Stiles The Univ. of Kansas Dept. of EECS 9/6/2005 Differential Line Vector Elements.doc 3/4 dr dxdy dz =++aaaˆˆˆ ddρρxyz d ρ d ρ ddρφcos ρφ sin ()ˆˆˆ()dz =++aaaxyz dddρρρ ˆˆ =+cosφφaaxy sin = aˆρ A change in position from coordinates ρ,,zφ to ρ +∆ρφ,,z results in a change in the position vector from r to r + ∆A. The vector ∆A is a directed distance extending from point ρ,,zφ to point ρ +∆ρφ,,z, and is equal to: d r ∆=∆A ρ dρ ˆˆ =∆ρφcosaaxy +∆ ρφ sin =∆ρ aˆρ y ∆A =∆ρˆap r x If ∆A is really small (i.e., as it approaches zero) we can define something called a differential displacement vector d A: Jim Stiles The Univ. of Kansas Dept. of EECS 9/6/2005 Differential Line Vector Elements.doc 4/4 d Alim ∆ A ∆→A 0 ⎛⎞dr = lim ⎜⎟∆A ∆→A 0 ⎝⎠d A ⎛⎞dr = ⎜⎟d A ⎝⎠d A For example: d r ddadρ ==ρρˆ d ρ ρ Essentially, the differential line vector d A is the tiny directed distance formed when a point changes its location by some tiny amount, resulting in a change of one coordinate value A by an equally tiny (i.e., differential) amount d A. The directed distance between the original location (at coordinate value A) and its new location (at coordinate value AA+ d ) is the differential displacement vector d A. d A r rd + A We will use the differential line vector when evaluating a line integral. Jim Stiles The Univ. of Kansas Dept. of EECS 9/6/2005 The Differential Line Vector for Coordinate Systems.doc 1/3 The Differential Displacement Vector for Coordinate Systems Let’s determine the differential displacement vectors for each coordinate of the Cartesian, cylindrical and spherical coordinate systems! Cartesian This is easy! ⎡⎤ dr⎛⎞ dxˆˆˆ⎛⎞dy ⎛⎞ dz dx== dx⎢⎥⎜⎟ axyz +⎜⎟ a + ⎜⎟ a dx dx⎣⎦⎝⎠ dx⎝⎠ dx ⎝⎠ dx ˆ =adxx ⎡⎤⎛⎞⎛⎞⎛⎞ dr dxˆˆˆdy dz dy== dy⎢⎥⎜⎟⎜⎟⎜⎟ axyz + a + a dy dy⎣⎦⎢⎥⎝⎠⎝⎠⎝⎠ dy dy dy ˆ =adyy ⎡⎤ dr⎛⎞ dxˆˆˆ⎛⎞dy ⎛⎞ dz dz== dz⎢⎥⎜⎟ axyz +⎜⎟ a + ⎜⎟ a dz dz⎣⎦⎝⎠ dz⎝⎠ dz ⎝⎠ dz ˆ =adzz Jim Stiles The Univ. of Kansas Dept. of EECS 9/6/2005 The Differential Line Vector for Coordinate Systems.doc 2/3 Cylindrical Likewise, recall from the last handout that: dadρ = ˆρ ρ Maria, look! I’m starting to see a trend! dr dx== dx aˆ dx dx x dr Q: It seems very apparent that: dy== dy aˆ dy dy y AA= ˆ dr dadA dz== dz aˆ dz dz z dr for all coordinates A; right ? ddadρ ==ρρˆ d ρ ρ A: NO!! Do not make this mistake! For example, consider d φ : φ = ˆ ρ φ d r Q: No!! dadφ ?!? ddφφ= φ How did the coordinate d ⎛⎞dxdy dz ρ get in there? =++ˆˆˆφ ⎜⎟aaadxyz ⎝⎠dddφφφ ⎛⎞dddzρφcos ρφ sin =++ˆˆˆφ ⎜⎟aaadxyz ⎝⎠dddφφφ ˆˆ =−()ρφsinacoadxy + ρ s φ φ ˆˆ ˆ =−()sinφρφaxy +co sφ a d =adφ ρ φ Jim Stiles The Univ. of Kansas Dept. of EECS 9/6/2005 The Differential Line Vector for Coordinate Systems.doc 3/3 The scalar differential value ρ d φ makes sense! The differential displacement vector is a directed distance, thus the units of its magnitude must be distance (e.g., meters, feet). The differential value dφ has units of radians, but the differential value ρ d φ does have units of distance. The differential displacement vectors for the cylindrical coordinate system is therefore: dr ddadρ ==ρρˆ d ρ p dr ddadφ ==φρφˆ d φ φ dr dz== dz aˆ dz dz z Likewise, for the spherical coordinate system, we find that: dr dr== dr aˆ dr dr r dr ddardθθ==ˆ θ d θ θ dr ddardφ ==φθφˆ sin d φ φ Jim Stiles The Univ. of Kansas Dept. of EECS 09/06/05 The Line Integral.doc 1/6 The Line Integral This integral is alternatively known as the contour integral. The reason is that the line integral involves integrating the projection of a vector field onto a specified contour C, e.g., ⋅ A ∫A(rdc ) C Some important things to note: * The integrand is a scalar function. * The integration is over one dimension. * The contour C is a line or curve through three- dimensional space. * The position vector rc denotes only those points that lie on contour C. Therefore, the value of this integral only depends on the value of vector field A()r at the points along this contour. Jim Stiles The Univ. of Kansas Dept. of EECS 09/06/05 The Line Integral.doc 2/6 Q: What is the differential vector d A, and how does it relate to contour C ? A: The differential vector d A is the tiny directed distance formed when a point moves a small distance along contour C. d A C r A Contour C c rdc + origin As a result, the differential line vector d A is always tangential to every point of the contour. In other words, the direction of d A always points “down” the contour. Q: So what does the scalar integrand A()rdc ⋅ A mean? What is it that we are actually integrating? A: Essentially, the line integral integrates (i.e., “adds up”) the values of a scalar component of vector field A()r at each and every point along contour C.