1 Administrative Notes: Final • Final Monday, 5/12, 7-10pm, DRES exam in 343 AH from 5-10pm. • Conflict Tuesday, 5/13, 8-11am • Room assignments via Moodle, by section • Double Conflicts 5/14, from 8-11am and 1:30-4:30pm. Please consult University Handbook for guidelines, to make sure you qualify.2 Evaluations and Office Hours • Evaluations on 5/7, Wednesday • Office Hours: • No office hours on 5/7 • Extended Office Hours on 5/9 and 5/12, 12-4pm, 371 AH • TA room, 343 AH, as usual + extra hours on Friday, 5/9, from 5-9pm.3 Summary We collect all higher-dimensional versions of the Fundamental Theorem of Calculus together here (without hypotheses) so that you can see more easily their essential similarity. Notice that in each case we have an integral of a “derivative” over a region on the left side, and the right side involves the values of the original function only on the boundary of the region.4 Summary cont’d5 Summary cont’d6 6 The Fundamental Theorem for Line Integrals If we think of the gradient vector ∇f of a function f of two or three variables as a sort of derivative of f, then the following theorem can be regarded as a version of the Fundamental Theorem for line integrals.7 7 Green's Theorem8 8 Green's Theorem Then Green’s Theorem gives the following formulas for the area of D:9 9 Curl If F = P i + Q j + R k is a vector field on and the partial derivatives of P, Q, and R all exist, then the curl of F is the vector field on defined by Let’s rewrite Equation 1 using operator notation. We introduce the vector differential operator ∇ (“del”) as10 10 Curl It has meaning when it operates on a scalar function to produce the gradient of f : If we think of ∇ as a vector with components ∂/∂x, ∂/∂y, and ∂/∂z, we can also consider the formal cross product of ∇ with the vector field F as follows:11 11 Curl So the easiest way to remember Definition 1 is by means of the symbolic expression12 12 Divergence If F = P i + Q j + R k is a vector field on and ∂P/∂x, ∂Q/∂y, and ∂R/∂z exist, then the divergence of F is the function of three variables defined by Observe that curl F is a vector field but div F is a scalar field.13 13 Divergence In terms of the gradient operator ∇ = (∂/∂x) i + (∂/∂y) j + (∂/∂z) k, the divergence of F can be written symbolically as the dot product of ∇ and F:14 14 Divergence If F is a vector field on , then curl F is also a vector field on . As such, we can compute its divergence. The next theorem shows that the result is 0.15 15 Vector Forms of Green’s Theorem The curl and divergence operators allow us to rewrite Green’s Theorem in versions that will be useful in our later work. We suppose that the plane region D, its boundary curve C, and the functions P and Q satisfy the hypotheses of Green’s Theorem. Then we consider the vector field F = P i + Q j.16 16 Vector Forms of Green’s Theorem Its line integral is and, regarding F as a vector field on with third component 0, we have17 17 Vector Forms of Green’s Theorem Therefore and we can now rewrite the equation in Green’s Theorem in the vector form18 18 Vector Forms of Green’s Theorem A second vector form of Green’s Theorem. This version says that the line integral of the normal component of F along C is equal to the double integral of the divergence of F over the region D enclosed by C.19 19 Surface Integrals of Vector Fields A surface integral of this form occurs frequently in physics, even when F is not ρ v, and is called the surface integral (or flux integral ) of F over S. In words, Definition 8 says that the surface integral of a vector field over S is equal to the surface integral of its normal component over S.20 20 Surface Integrals of Vector Fields If S is given by a vector function r(u, v), then n is given by Equation 6, and from Definition 8 and Equation 2 we have where D is the parameter domain. Thus we have21 21 Stokes' Theorem . The orientation of S induces the positive orientation of the boundary curve C. This means that if you walk in the positive direction around C with your head pointing in the direction of n, then the surface will always be on your left22 22 The Divergence Theorem The boundary of E is a closed surface, and we use the convention, that the positive orientation is outward; that is, the unit normal vector n is directed outward from E. Thus the Divergence Theorem states that, under the given conditions, the flux of F across the boundary surface of E is equal to the triple integral of the divergence of F over
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