548Mathcad - Mathcad Resources Gradient, Divergence, And Curl

CALCULUS AND DIFFERENTIAL EQUATIONS quicksheets Gradient, Divergence, and Curl

This QuickSheet illustrates how to define and evaluate the vector operators gradient, divergence, and curl. Gradient

The gradient of a scalar-valued function

 x0    f ( x) = f  x1  x   2 is defined by

∂  f ( x)    ∂x0    ∂ f ( x)  ∇ x f ( x) =   ∂x1    ∂  f ( x )    ∂x2  You can compute the gradient using the Mathcad gradient operator. For example, suppose you define the following scalar-valued function: f ( x) := ( x0) 2 ⋅ x1 ⋅ ( x2) 3 To insert the gradient operator, press [Ctrl] [Shift] G: ∇

Type the variable vector x in the lower placeholder, and type the function f(x) in the upper placeholder. ∇ x f ( x) Finally, type [Ctrl] [.] to insert the symbolic equal sign and press [Enter].

 2 ⋅ x0 ⋅ x1 ⋅ ( x2) 3     ∇ x f ( x) →  ( x0) 2 ⋅ ( x2) 3   3 ⋅ x 2 ⋅ x ⋅ x 2  ( 0) 1 ( 2)  If you define the vector x numerically, the gradient is a numerical vector:

1  x :=  3     −1   −6  ∇ x f ( x) =  − 1    9  Divergence

The divergence of a vector-valued function A(x,y,z) = (A(x,y,z)0, A(x,y,z)1, A(x,y,z)2) is defined by div ( A , x , y , z) :=

d d d A ( x , y , z) 0 + A ( x , y , z) 1 + A ( x , y , z) 2 dx dy dz

This can also be written as div = ∇ ⋅ A where ∇ = 

d  dx

d dy

d   is the "del" operator. dz 

For example, if you define

x := x

 x⋅z    A( x , y , z) :=  −y2   2  2⋅x ⋅y you can evaluate the divergence symbolically by: div ( A , x , y , z) → z − 2 ⋅ y After substituting values for x, y, and z, you can evaluate the divergence numerically: div ( A , 1 , 1 , 1) = −1 Curl

The curl of a vector-valued function A is defined by

d   A ( x , y , z) 2 − d A ( x , y , z) 1  dz  dy  d  d curl ( A , x , y , z) :=  A( x , y , z) 0 − A( x , y , z) 2  dx  dz  d  d  dx A ( x , y , z ) 1 − dy A ( x , y , z ) 0    In terms of the del operator, this can be written as curl = ∇ × A

To symbolically evaluate the curl of the function A defined previously, use the symbolic equal sign:

 2 ⋅ x2    curl ( A , x , y , z) →  x − 4 ⋅ x ⋅ y    0   Or, you can define values for x, y, and z, and evaluate the curl numerically: 2  curl ( A , 1 , 1 , 1) =  −3 

  0 

Here are some more examples: f ( x , y , z) := x ⋅ y ⋅ z 2

3

B ( x , y , z) := f ( x , y , z) ⋅ A( x , y , z)

 x 3 ⋅ y ⋅ z 4  B ( x , y , z ) →  − x 2 ⋅ y3 ⋅ z 3     2 ⋅ x 4 ⋅ y2 ⋅ z 3    div ( B , x , y , z) → 6 ⋅ x 4 ⋅ y2 ⋅ z 2 − 3 ⋅ x 2 ⋅ y2 ⋅ z 3 + 3 ⋅ x 2 ⋅ y ⋅ z 4

 4 ⋅ x 4 ⋅ y ⋅ z 3 + 3 ⋅ x 2 ⋅ y3 ⋅ z 2  curl ( B , x , y , z) →  4 ⋅ x 3 ⋅ y ⋅ z 3 − 8 ⋅ x 3 ⋅ y2 ⋅ z 3     − x 3 ⋅ z 4 − 2 ⋅ x ⋅ y3 ⋅ z 3   