MIT 3.016 Fall 2005 � W.C Carter Lecture 13 c80 Oct. 17 2005: Lecture 13: Differential Operations on Vectors Reading: Kreyszig Sections: 8.10 (pp:453–56) , 8.11 (pp:457–459) § §Generalizing the Derivative The number of different ideas, whether from physical science or other disciplines, that can be understood with reference to the “meaning” of a derivative from the calculus of scalar functions is very very large. Our ideas about many topics, such as price elasticity, strain, stability, and optimization, are connected to our understanding of a derivative. In vector calculus, there are generalizations to the derivative from basic calculus that acts on a scalar and gives another scalar back: gradient (�): A derivative on a scalar that gives a vector. curl (�×): A derivative on a vector that gives another vector. divergence (�·): A derivative on a vector that gives scalar. Each of these have “meanings” that can be applied to a broad class of problems. The gradient operation on f(�x) = f(x, y, z) = f(x1, x2, x3), � ∂f ∂f ∂f � � ∂ ∂ ∂ �gradf = �f , , = , , f (13-1)∂x ∂y ∂z ∂x ∂y ∂z has been discussed previously. The curl and divergence will be discussed below. Mathematica r� Example: Lecture-13 Gradient of a several 1/r potentials Three Electric Charges Divergence and Its Interpretation qcqckCoordinate Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The above definitions are for a Cartesian (x, y, z) system. Sometimes it is more convenient to� MIT 3.016 Fall 2005 � W.C Carter Lecture 13 c81 work in other (spherical, cylindrical, etc) coordinate systems. In other coordinate systems, the derivative operations �, �·, and �× have different forms. These other forms can be derived, or looked up in a mathematical handbook, or specified by using the Mathematica r� package “VectorAnalysis.” Mathematica r� Example: Lecture-13 Coordinate System Transformations Converting between Cartesian and Spherical Coordinates with Mathematica rThe divergence operates on a vector field that is a function of position, �v(x, y, z) = �v(�x) = (v1(�x), v2(�x), v3(�x)), and returns a scalar that is a function of position. The scalar field is often called the divergence field of �v or simply the divergence of �v. ∂v1 ∂v2 ∂v3 � ∂ ∂ ∂ � � ∂ ∂ ∂ �div �v(�x) = � · �v = + + = , , · (v1, v2, v3) = , , �v (13-2)∂x ∂y ∂z ∂x ∂y ∂z ∂x ∂y ∂z · Think about what the divergence means, Curl and Its Interpretation The curl is the vector valued derivative of a vector function. As illustrated below, its operation can be geometrically interpreted as the rotation of a field about a point. For a vector-valued function of (x, y, z): �v(x, y, z) = �v(�x) = (v1(�x), v2(�x), v3(�x)) = v1(x, y, z)ˆi + v2(x, y, z)ˆj + v3(x, y, z)ˆ(13-3)k� × �� � � × ��� �MIT 3.016 Fall 2005 � W.C Carter Lecture 13 c82 the curl derivative operation is another vector defined by: �� ∂v3 � ∂v1 � ∂v2 ∂v1 ��∂v2 ∂v3curl �v = v = (13-4)∂y − ∂z − ∂x −, ,∂z ∂x ∂y or with the memory-device: curl �v = v = det ⎛⎝ ˆˆi ˆj k ∂ ∂ ∂ ∂x ∂y ∂z v1 v2 v3 ⎞⎠ (13-5) Mathematica r� Example: Lecture-13 Calculating the Curl of a Function n 2 Consider the vector function that is often used in Brakke’s Surface Evolver program: n 2 nzw = (yˆi − xˆj)(x2 + y2)(x2 + y2 + z2) This can be shown easily, using Mathematica r� , to have the property: nzn−1 w = 2)1+ (xˆi + yˆj + zk)� × �(x2 + y2 + z ˆwhich is spherically symmetric for n = 1 and convenient for turning surface integrals over a portion of a sphere into a path-integral over a curve on a sphere. 1. Create vector function �w above and visualize using the PlotVectorField3D func-tion in Mathematica r� ’s PlotField3D package. 2. The function will be singular for n > 1 along the z − axis, this singularity will b e communicated during the numerical evaluations for visualization unless some care is applied. 3. Demonstrate the above assertion about �w and its curl. 4. Visualize the curl: note that the field is points up with large magnitude near the vortex at the origin. 5. Demonstrate that the divergence of the curl of �w vanishes for any n. One important result that has physical implications is that a the curl of a gradient is always zero: f(�x) = f(x, y, z): � × (�f) = 0 (13-6) Therefore if some vector function �F (x, y, z) = (Fx, Fy, Fz) can be derived from a scalar poten-tial, �f = F , then the curl of F must be zero. This is the property of an exact differentialikj∂vy∂x>0∂vx∂y<0MIT 3.016 Fall 2005 � W.C Carter Lecture 13 c83 df = (�f) · (dx, dy, dz) = �· (dx, dy, dz). Maxwell’s relations follow from equation 13-6: F 0 = ∂Fz ∂y − ∂Fy ∂z = ∂ ∂f ∂z ∂y − ∂ ∂f ∂y ∂z = ∂2f ∂z∂y − ∂2f ∂y∂z 0 = ∂Fx ∂z − ∂Fz ∂x = ∂ ∂f ∂x ∂z − ∂ ∂f ∂z ∂x = ∂2f ∂x∂z − ∂2f ∂z∂x (13-7) 0 = ∂Fy ∂x − ∂Fx ∂y = ∂ ∂f ∂y ∂x − ∂ ∂f ∂x ∂y = ∂2f ∂y∂x − ∂2f ∂x∂y Another interpretation is that gradient fields are curl free, irrotational, or conservative. The notion of conservative means that, if a vector function can be derived as the gradient of a scalar potential, then integrals of the vector function over any path is zero for a closed curve—meaning that there is no change in “state;” energy is a common state function. Here is a picture that helps visualize why the curl invokes names associated with spinning, rotation, etc. Figure 13-1: Consider a small paddle wheel placed in a set of stream lines defined by a vector field of position. If the vy component is an increasing function of x, this tends to make the paddle wheel want to spin (positive, counter-clockwise) about the ˆk-axis. If the vx component is a decreasing function of y, this tends to make the paddle wheel want to spin (positive, counter-clockwise) about the ˆk-axis. The net impulse to spin around the ˆk-axis is the sum of the two. Note that this is independent of the reference frame because a constant velocity �v = const.