We like to depict a vector field by drawing the outputs as vectors with their tails at the input (see below).

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Patrick Wade
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1 Math 55 - Vector Calculus II Notes 4. Vector Fields A function F is a vector field on a subset S of R n if F is a function from S to R n. particular, this means that F(x, x,..., x n ) = f (x, x,..., x n ), f (x, x,..., x n ),..., f n (x, x,..., x n ), for some scalar-valued functions f, f,..., f n defined on S. In We like to depict a vector field by drawing the outputs as vectors with their tails at the input (see below). Examples: () F(x, y) = x, y is a vector field on R. This is an example of a radial vector field. Below is a picture. Note: F(x, y) = x + y. So the output vector s length at an input is equal to the input s distance to the origin

2 () F(x, y) = y, x is a vector field on R. This is an example of a rotational vector field. Below is a picture. Note: F(x, y) = x + y. So the output vector s length at an input is equal to the input s distance to the origin

3 (3) F(x, y) = y, is a vector field on R. This is an example of a shear vector field. Below is a picture. Note: F(x, y) = y. So the output vector s length at an input is equal to the input s distance to the x-axis

4 (4) F(x, y, z) =,, x + y is a vector field on R 3. This is an example of a flow vector field. Below is a picture.

5 (5) F(x, y) = x y, is a vector field on {(x, y) R y > } (the upper half of R ). y This is an example of a gradient vector field. Below is a picture

6 Basic Types of Vector Fields: Let r = x, x,..., x n. A radial vector field is one of the form F(x, x,..., x n ) = f(x, x,..., x n )r where f is a scalar-valued function on R n. We can express such a field in the form F(x, x,..., x n ) = f(x, x,..., x n ) r ˆr, where ˆr = r r is a unit vector (so f(x, x,..., x n ) r is the magnitude). In particular, we will often use radial vector fields of the form F(x, x,..., x n ) = p is a real number. See example. r r p where A vector field F on a subset S of R n is a gradient vector field if there exists a scalar-valued function φ on S such that F = φ. We call such a function φ a potential function. See example 5. Determine a potential function! Level sets of a potential function φ are called equipotential sets. In the case where n =, these are level curves and are called equipotential curves. In the case where n = 3, these are level surfaces of a potential function φ are called equipotential surfaces. Recall that the gradient evaluated at a point P is always orthogonal to the level set (curve, surface, etc.) at P (in the direction of maximum increase). So a gradient vector field is everywhere orthogonal to equipotential sets! A directed curve through a gradient vector field that is everywhere orthogonal to equipotential sets is a flow curve or streamline.

7 Consider the vector field F(x, y) = x,. This is a gradient vector field. One potential function is φ(x, y) = x + y. Here are some equipotential curves in black and a flow curve in red (shown within the vector field): Let s verify that the gradient vector field is orthogonal to its equipotential curves at any point (x, y). Since the level curve looks like φ(x, y) = k for some constant k we can use the rule y = φ x φ y to get y = x = x. Then T(x, y) =, x is tangent to the equipotential curve at (x, y). Since F(x, y) = x, we have that F(x, y) T(x, y) =, which shows F is orthogonal to its equipotential curves.

8 Determine where the gradient vector field F = y, x (often out of laziness we write F for F(x, y)) is tangent to and normal to the curve C parameterized by r(t) = t, t. Let (x, x ) be an arbitrary point on the curve. A tangent vector is T =, x. Also F(x, x ) = x, x. F is tangent to C wherever x, x = c, x for some non-zero scalar c: Then c = x and x = cx. From the second equation, either x = or c = /. 3 If x = then c =. If c = / and x = ±. ( ) 3 So F is tangent to curve C at ±, and (, ). F is normal to the curve wherever F(x, x ) T = : So we solve x, x, x =, which is equivalent to x x =. ( ) Hence x = ±. So F is normal to the curve at the ± 3 3,. 3 The picture confirms our calculations:

9 Now let s construct a vector field with some desired properties. Let s find a non-constant vector field on R (excluding the origin) that is always normal to the line y = x and has unit magnitude at every point (excluding the origin). Let F = f(x, y), g(x, y). Since T =, is tangent to the line y = x we require that f(x, x) + g(x, x) =. The other requirement is that f(x, y) + g(x, y) =. There are lots y x of possibilities, but one is f(x, y) = and g(x, y) = x + y x + y. Here is this vector field: The solution above isn t unique, here is another solution based off of normalizing F = x y, x :

10 Applications: -The electric field E in space due to a point charge at (,, ) satisfies E = V where k V (x, y, z) = and k > is a constant. x + y + z Let s show that the radial component of the electric field is k/r where r = x + y + z. E = k (x + y + z ) x, y, z = k 3/ x + y + z ˆr where ˆr = x, y, z. x + y + z Hence the radial component of E is k/r where r = x + y + z. -A flow curve to F = f(x, y), g(x, y) satisfies y = Find equations for the flow curves of F = + x, y. g(x, y) f(x, y). y = y + x y y = + x y = tan (x) C y = Here is the flow curve with C = : C tan (x)

4 to find the dimensions of the rectangle that have the maximum area. 2y A =?? f(x, y) = (2x)(2y) = 4xy

4 to find the dimensions of the rectangle that have the maximum area. 2y A =?? f(x, y) = (2x)(2y) = 4xy Optimization Constrained optimization and Lagrange multipliers Constrained optimization is what it sounds like - the problem of finding a maximum or minimum value (optimization), subject to some other