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Week 2 Vector Operators Divergence and Stoke’s Theorems

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Gradient Operator The gradient is a vector operator denoted and sometimes also called “del.” It is most often applied to a real function of three variables. In Cartesian coordinates, the gradient of f(x, y, z) is given by grad (f) = f = x ∂f/∂x + y ∂f/∂ + z ∂f/∂z The expression for the gradient in cylindrical and spherical coordinates can be found on the inside back cover of your textbook.

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Significance of Gradient The direction of grad(f) is the orientation in which the directional derivative has the largest value and |grad(f)| is the value of that directional derivative. Furthermore, if grad(f) ≠ 0, then the gradient is perpendicular to the “level” curve z = f(x,y)

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Example As an example consider the gravitational potential on the surface of the Earth: V(z) = -gzwhere z is the height The gradient of V would be V = z ∂V/∂z = -g a z

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Exercise Consider the gradient represented by the field of blue arrows. Draw level curves normal to the field.

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Exercise Calculate the gradient of f = x 2 + y 2 f = 2xy f = e x sin y

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Exercise Consider the surface z 2 = 4(x 2 + y 2 ). Find a unit vector that is normal to the surface at P:(1, 0, 2).

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Laplacian Operator The Laplacian of a scalar function f(x, y, z) is a scalar differential operator defined by 2 f = [∂ 2 /∂x 2 + ∂ 2 /∂y 2 + ∂ 2 /∂z 2 ]f The expression for the Laplacian operator in cylindrical and spherical coordinates can be found in the back cover of your textbook. The Laplacian of a vector A is a vector.

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Applications The Laplacian quite important in electromagnetic field theory: It appears in Laplace's equation 2 f = 0 the Helmholtz differential equation 2 f + k 2 f = 0 and the wave equation 2 f = (1/c) 2 ∂ 2 f/∂x 2

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Exercise Calculate the Laplacian of: f = sin 0.1πx f = xyz f = cos( k x x ) cos( k y y ) sin( k z z )

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Curl Operator The curl is a vector operator that describes the rotation of a vector field F: x F At every point in the field, the curl is represented by a vector. The direction of the curl is the axis of rotation, as determined by the right-hand rule. The magnitude of the curl is the magnitude of rotation.

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Definition of Curl where the right side is a line integral around an infinitesimal region of area A that is allowed to shrink to zero via a limiting process and n is the unit normal vector to this region.

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Line Integral A line integral is an integral where the function is evaluated along a predetermined curve.

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Significance of Curl The physical significance of the curl of a vector field is the amount of "rotation" or angular momentum of the contents of given region of space.

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Exercise Consider the field shown here. If we stick a paddle wheel in the first quadrant would it rotate? If so, in which direction?

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Curl in Cartesian Coordinates In practice, the curl is computed as The expression for the curl in cylindrical and spherical coordinates can be found on the inside back cover of your textbook.

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Exercise Find the curl of F = x a x + yz a y – (x 2 + z 2 ) a z.

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Divergence Operator The divergence is a vector operator that describes the extent to which there is more “flux” exiting an infinitesimal region of space than entering it: · F At every point in the field, the divergence is represented by a scalar.

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Definition of Divergence where the surface integral is over a closed infinitesimal boundary surface A surrounding a volume element V, which is taken to size zero using a limiting process.

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Surface Integral It’s the integral of a function f(x,y,z) taken over a surface.

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Example Consider a field F = F o /r 2 a r. Show that the ratio of the flux coming out of a spherical surface of radius r=a to the volume of the same sphere is = 3F o /4a 3 First calculate = 4 π F o Then calculate V = 4π a 3 /3

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Significance of Divergence The divergence of a field is the extent to which the vector field flow behaves like a source at a given point.

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Divergence in Cartesian Coordinates In practice the divergence is computed as The expression for the divergence in cylindrical and spherical coordinates can be found on the inside back cover of your textbook.

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Exercise Determine the following: divergence of F = 2x a x + 2y a y. divergence of the curl of F = 2x a x + 2y a y.

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Divergence Theorem The volume integral of the divergence of F is equal to the flux coming out of the surface A enclosing the selected volume V : The divergence theorem transforms the volume integral of the divergence into a surface integral of the net outward flux through a closed surface surrounding the volume.

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Example Consider the “finite volume” electric charge shown here. The divergence theorem can be used to calculate the net flux outward and the amount of charge in the volume. Requirement: the field must be continuous in the volume enclosed by the surface considered.

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Exercise Consider a spherical surface of radius r = b and a field F = (r/3) a r. Show that the divergence of F is 1. Show that the volume integral of the divergence is (4π/3) b 3 Show that the flux of F coming out of the spherical surface is (4π/3) b 3

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Stokes' Theorem It states that the area integral of the curl of F over a surface A is equal to the closed line integral of F over the path C that encloses A: Stoke’s Theorem transforms the circulation of the field into a line integral of the field over the contour that bounds the surface.

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Significance of Stoke’s Theorem The integral is a sum of circulation differentials. The circulation differential is defined as the dot product of the curl and the surface area differential over which it is measured.

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Exercise Consider the rectangular surface shown below. Let F = y a x + x a y. Verify Stoke’s Theorem. A B

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Homework Read book sections 3-3, 3-4, 3-5, 3-6, and 3-7. Solve end-of-chapter problems 3.32, 3.35, 3.49, 3.39, 3.41, 3.43, 3.45, 3.48

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