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Ch14Notes - Chapter 14 VECTOR CALCULUS Lecture Notes...

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Chapter 14 VECTOR CALCULUS - Lecture Notes Section 14.1 Vector Fields A function F that associates a vector with each point in space is called a Vector Field . A function f that associates a number with each point in space is called a Scalar Field . Examples: Vector Fields Scalar Fields Electric Fields ( Coulomb’s Law ) Temperature for each Point Magnetic Fields Distance from the origin for each point Force Fields Pressure for each point for some region Gravitational Fields Density for each point for some region Velocity Fields (flow of air over a car or plane, flow of water through a pipe) We will be working with Steady State Fields . These are fields that are independent of time they remain constant. Also these fields don’t have to be de fined on the entire 3D space but can be defined on a certain set of points in space that might represent a curve, surface, or solid. Newton’s Law of Gravitation: The magnitude of the force of attraction between objects M and m is given by where d is the distance between the objects. G is our gravitational constant. Assume M is at the origin and is the position vector for m. The distance point m is from the origin is given by which is equivalent to . So the magnitude of the gravitational force between M and m is given by: Obviously the vector should point toward the origin. We know points away from the origin toward the object m so would point toward the origin. The unit vector would be . Thus the vector function describing the gravitational force acting on m is: This is called a central force field since the force vector points toward the origin at each point. Now assume GMm = c which is a constant. This simplifies are vector function to Let is a scalar field. This is what we worked with in chapter 11. I could write for a more compact notation of the function.
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Notice A vector field F that is the gradient of a scalar field f is called a Conservative Vector Field and f is called its potential function . What condition must be met in order for a vector field to have a potential function? Let Thus we must have: We can use this condition to find the potential function. But how do we know there is an f that will satisfy this condition? Notice So a potential function exists only if: . Check this first, then move on to finding the potential function. Example: Let First check So our potential function is: Double Check: Thus Conservative Vector Field
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There is more than one way to find the potential function. After we found: We could then take the derivative of this with respect to y to get: Set this equal to N(x,y) since Thus The advantage of this way is that you only have to integrate once. Example: Let First check Thus Another perspective of the Potential Function: Let’s work backward: Let y(x) be a solution to a first -order differential equation where y is a function of x and is implicitly defined by: Differentiate each side with respect to x: Thus the differential equation is: And we know the solution to this differential equation is: Let’s show our potential function from our last example satisfies the differential equation: Should satisfy the differential equation: x y x f
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