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Unformatted text preview: dA. Remark: Before using Green’s Theorem, you must be sure: • P and Q have CONTINUOUS partial derivatives on D ; • ∂D is a closed curve with POSITIVE orientation. Positive Orientation
The orientation of ∂D in Green’s Theorem is such that, as one traverses along the boundary in this direction, the region D is always on the left hand side. We call this the positive orientation of the boundary. Explanation Suppose you are driving or taking a taxi to Engineering. To go to your destination, the car must in the direction as −→. If you car is in the direction as , you need to turn around, and change to the direction −→ ﬁnally. So, the correct direction makes the destination on the LEFT HAND SIDE. 192 Summarization of Line Integral
Suppose we are given a function, and want to evaluate its line integral along the curve C . 10 c 8 6 4 2 c*
5 10 15 For diﬀerent types of functions, we have diﬀerent methods: • For Scalar Field f , we can direct compute the line integral by the formula:
b f (x, y, z )ds =
C a f x(t), y (t), z (t) r (t) dt. • For Vector Field F, we have three methods: – Direct Computation: F • dr =
b a C F r(t) • r (t)dt. – Fundamental Theorem: if we can ﬁnd an f , such that F = ∇f , then we can use this method.
B A F • dr = f (B ) − f (A). – Green’s Theorem (2-Dim Case) or Stokes’ Theorem (3-Dim Case): F(x, y ) =⇒ F(x, y, z ) =⇒ Remark: ∗ The orientation should be POSITIVE. If not, just add a “-” before the integral. ∗ For the two cases, C should be a CLOSED curve. If C is not closed as in the above graph, then
C C F • dr = F • dr = R (Qx − Py ) dA curlF • dS (Double Integral) (Surface Integral) C S F • dr = C +C ∗ F • dr − C∗ F • dr 193 Solution (1) Direct Computation: Here, C is a piecewise smooth curves made up of 4 straight lines C1 , C2 , C3 , C4 , and the direction of C is anti-clockwise To parametrize the four line segments, we notice that each one is parallel to x-axis or y -axis, that means when moving along C , one of the coordinates (x or y ) is constant, and so we can let the other coordinate to be the parameter t. • For C1 in the following graph, it is obvious to see that y = 0 is constant, and 0 ≤ x ≤ 2, so we deﬁne the parameter equations by x = t, y = 0, 0 ≤ t ≤ 2. When t moves from 0 to 2, it is ANTI-CLOCKWISE. • For C2 in the following graph, it is obvious to see that x = 2 is constant, and 0 ≤ y ≤ 3, so we deﬁne the parameter equations by x = 2, y = t, 0 ≤ t ≤ 3. When t moves from 0 to 3, it is ANTI-CLOCKWISE. • For C3 in the following graph, it is obvious to see that y = 3 is constant, and 0 ≤ x ≤ 2, so we deﬁne the parameter equations by x = t, y = 3, 0 ≤ t ≤ 2. When t mo...
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- Spring '09