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Unformatted text preview: me ∇f ) are called conservative ﬁelds in Physics. For example, if there is no external force (for example, no friction force here), then the total energy (including kinetic energy and potential energy) of an object is conserved when moving from A to B . Recall: Field
In Physics, the notation ﬁeld stands for a multi-variable function. So, we have learnt two kinds of ﬁelds: scalar ﬁeld and vector ﬁeld. Check tutorial notes 8 for more details. Recall: Conservative Field
We have many ways to deﬁne the conservative ﬁeld, and all the deﬁnitions are equivalent. Here is the way used in Thomas’ Calculus: DEFINITIONS Path Independence, Conservative Field Let F be a field defined on an open region D in space, and suppose that for any B two points A and B in D the work 1A F # dr done in moving from A to B is the same over all paths from A to B. Then the integral 1 F # dr is path independent in D and the field F is conservative on D.
An alternative deﬁnition is: A ﬁeld F is conservative if and only if it is the gradient ﬁeld of a scalar function f ; that is, if and only if for F = ∇f for some f . The function f has a special name: potential function. 178 Remark: 1. Please note that the conservative function is F, not the potential function f . 2. Revisit tutorial 8 to check the criteria of conservative ﬁelds (Component Test), and the method to get the potential function f . F is called a conservative ﬁeld if there is a function f such that F = ∇f . In this case, f is called a potential function for F. ⎞ ⎛ fx ⎟ ⎜ F = ∇f = ⎝ f y ⎠ fz Here are some properties of Conservative Field and its Potential Function f : (a) Component Test. (Check details in tutorial 8.) If F has three components, F = M (x, y, z )i + N (x, y, z )j + P (x, y, z )k, then the component test says: if ∂M ∂N = , ∂x ∂y then F is conservative. Remark: This is the most useful way to determine a vector ﬁeld is conservative. ∂P ∂M = , ∂x ∂z ∂P ∂N = , ∂y ∂z (b) (Path Independent) We line integrate the function along any curve will get the same answer.
B A ∇f • dr = f (B ) − f (A). (c) We line integrate the function along any closed curve will get 0: f ds = 0.
C (d) If we can change the order of second-order partial derivatives of the potential function freely. (Check it in tutorial 8.) fxy = fyx , fxz = fzx , fyz = fzy . 179 Example: Non-Conservative Field in Singapore Suppose you take a taxi from NUS to go to Jurong Easy, your routine is NUS =⇒ Clementi =⇒ Jurong East, then the fee is below S$ 10. However, if you choose the routine: Orchard =⇒ Hougang =⇒ Woodlands =⇒ Chao Chu Kang =⇒ Jurong East, then, the fee is much more expensive. That means the fee for taxi is path dependent (Not Conservative). 180 Example: Conservative Field in Singapore Suppose you take MRT to go to Jurong East from Clementi, then you have t...
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This note was uploaded on 05/05/2009 for the course MA MA1505 taught by Professor Ma1505 during the Spring '09 term at National University of Juridical Sciences.
- Spring '09