9.4 Vector and Scalar Fields; Derivatives

Transcription

1 9.4 Vector and Scalar Fields; Derivatives Vector fields A vector field v is a vector-valued function defined on some domain of R 2 or R 3. Thus if D is a subset of R 3, a vector field v with domain D associates to each point P in D a vector: v (p) = [v1 (p), v 2 (p), v 3 (p)] Scalar fields A scalar field f is a real-valued function defined on some domain of R 2 or R 3. Thus if D is a subset of R 3, a scalar field f with domain D associates to each point P in D a number f (p). Of course a vector field might also be two-dimensional, in which case it is of the form v (p) = [v1 (p), v 2 (p)]. 1 / 14

2 Examples of vector fields 1. Force fields Let D be some domain in R 3. Let F denote a force acting on each point of D. Then F is a vector field which we call a force field. Exercise. Place mass M at a point P 0 = (x 0, y 0, z 0 ). Then this mass exerts an attracting gravitational force on any mass m placed at some other point Q = (x, y, z) in R 3. This gravitational force F is proportional to the product of the masses and inversely proportional to the square of the distances between the masses. Write down in vector form and component form how much is F (Q). 2 / 14

3 Examples of vector fields 2. Velocity fields If we have a fluid in R 3 (could be movement of water or a blowing wind) and v (P) is the vector field which is the velocity of the fluid at point P. Or we might have an object moving in space in some way and we wish to study its velocity at each point P in space. That gives rise to the velocity vector field v (P). Exercise. Say we have a fluid in R 3 which swirls around the z-axis with constant angular velocity ω radians/sec. Write down the velocity field v (x, y, z) = [v 1 (x, y, z), v 2 (x, y, z), v 3 (x, y, z)] which gives the velocity of the fluid at the point (x, y, z). Make use of relevant results we derived on vector products in the previous section. 3 / 14

4 Examples of vector fields Let C be a curve in R 2 or R Tangent fields Assume that C is oriented in some way and imagine a particle moving along C in the direction prescribed by the orientation. At each point P of C let T (P) be a unit tangent vector to the curve at P, assumed to be pointing in the direction given by the orientation. Then T is a vector field which we call the tangent field. 4 / 14

5 Examples of vector fields 4. Normal fields In any of the following examples, N represents a vector field known as a normal field. Let S be a closed surface in R 3, such as for example a sphere. At each point P of the surface, let N (P) denote the outward pointing unit normal vector to the surface. Or S might be a surface bounded by a curve (for example a hemishphere). We could then take N (P) to be the upward pointing unit normal vector at point P of the boundary of S. Or S might be the region in R 2 enclosed by a simple closed curve C. Then for each point P on S we could take N (P) to be the unit normal vector at P which points out of the region S. 5 / 14

6 Examples of scalar fields Some examples of scalar fields For a given region T in R 2 or R 3, all of the following functions f : T R defines a scalar field on T. 1 f (p) is one of the components of a vector field on T. 2 If a certain region T is heated in some way, let f (P) denote the temperature at point P of T. 3 Let P 0 be a fixed point in T. For any P in T, let f (P) be the distance of P to P 0. 4 (Potential function of a conservative vector field) Let F be a vector field on T. We say that F is conservative if the work done by the force in moving a particle between any two points of T is independent of which path we take within T, i.e. it depends only on the initial point and the endpoint of that path. Fix a point P 0 in T. Define f (P) to be the work done by F in moving a particle from P 0 to P. Then f is a scalar field on T, and we refer to f as a potential of F. We call the quantity f (P) f (P 0) the potential difference between P and P 0. 6 / 14

7 Pictures associated with vector and scalar fields Picture associated with a vector field Let F be a vector field on some domain T. At each point P in the domain, draw the vector F (P) as an arrow with initial point P. If we do this for enough points P, we can get a sense of how the field acts. Exercise. Sketch each of the following 2-dimensional vector fields. a) F = ı + 2 j b) F = x j c) F = [ y, x]. Do you recognize this vector field from an earlier exercise? 7 / 14

8 Pictures associated with vector and scalar fields Picture associated with a scalar field Let f (x, y) be a scalar field on some domain T. For each choice of a constant c, we call the set a level curve of f. {(x, y) T : f (x, y) = c} If we plot on the same sketch several different such level curves, we can get a sense of how the scalar field behaves. If we view f as representing temperature, then the level curves are referred to as isotherms. If we view f as the potential for a certain conservative force field, then the level curves are referred to as equipotential lines. If instead f (x, y, z) is a scalar field of three variables for (x, y, z) in a domain T in R 3, then the set {(x, y, z) T : f (x, y, z) = c} is now a surface in R 3, called a level surface of f. 8 / 14

9 Pictures associated with vector and scalar fields Exercise. 1 Identify and sketch the equipotential lines for each of the following potentials. a) f (x, y) = xy y b) f (x, y) = x 2 + y 2 2 Identify the level surfaces of the following scalar functions. a) f (x, y, z) = x y 2. b) f (x, y, z) = x 2 + y 2 z 2 9 / 14

10 Vector Calculus Let v (t) be a vector-valued function of one variable: We define the derivative v (t) in the obvious way: v (t) = [v1 (t), v 2 (t), v 3 (t)]. v (t) := lim h 0 v (t + h) v (t) h Vector subtraction and multiplation by the scalar 1/h produces vectors, so each expression is a vector. v (t + h) v (t) Since we can measure distance between two vectors, the meaning of the limit is that v (t) is the unique vector with the property that v (t + h) v (t) v (t) h 0 as h 0. h 10 / 14

11 Vector Calculus Recall that for vector [a, b, c], we ve defined [a, b, c] = a 2 + b 2 + c 2. It follows that the following all hold: a, b, c [a, b, c] [a, b, c] = a 2 + b 2 + c 2 a + b + c. Thus if we have a sequence of vectors [a n, b n, c n ], then it follows from the above inequalities that [a n, b n, c n ] 0 as n a n 0, b n 0, and c n 0 as n. From this we deduce that v (t) = [v 1(t), v 2(t), v 3(t)]. We have similar results for vector fields F (x, y, z) = [F 1 (x, y, z), F 2 (x, y, z), F 3 (x, y, z)]. For example the partial derivative F x is calculated by [ F x = F1 x, F 2 x, F ] 3 x with similar formulas for partials of F with respect to y or z. 11 / 14

12 Vector Calculus Each of the following formulas can be proven: 1 (Linearity) ( u + v ) = u + v (c u ) = c u Algebraic Properties (for one variable fields) 2 (Product rules) Let u and v be vector-valued functions of one variable, and let f (t) be a scalar field. ( u v ) = u v + u v ( u v ) = u v + u v ( f (t) v (t) ) = f (t) v (t) + f (t) v (t) 12 / 14

13 Vector Calculus Exercise. 1 If v (t) = [2t, 3t 2, cos t], calculate v (t). 2 If F = [xy, z 2 x 3, x 2 + y 2 + z 2 ], calculate each of the first partial derivatives of F. 13 / 14

14 Vector Calculus Exercise. Let r (t) be a vector-valued function with the property that there exists a constant c such that for all t we have r (t) = c. Explain why it must be the case that r (t) r (t) for all t. 14 / 14