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cvgmt
cvgmt
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You can use .5 {-y, x}/(x^2 + y^2) // TraditionalForm to see that .5 {-y, x}/(x^2 + y^2) can be use to express the vector value function.

$$\left(\frac{-0.5y}{x^2+y^2},\frac{0.5x}{x^2+y^2}\right)$$

and actually,{-y, x}/(x^2 + y^2) is the gradient of the function ArcTan[y/x] or ArcTan[x,y]

Grad[ArcTan[x, y], {x, y}]
Grad[ArcTan[y/x], {x, y}] // Simplify
(* {-(y/(x^2 + y^2)), x/(x^2 + y^2)} *)


GraphicsRow[{VectorPlot[{-(y/(x^2 + y^2)), x/(
    x^2 + y^2)}, {x, y} \[Element]∈ Disk[]], 
  VectorPlot[
   Grad[ArcTan[x, y], {x, y}] // Evaluate, {x, y} \[Element]∈ Disk[]]}]

enter image description here

You can use .5 {-y, x}/(x^2 + y^2) // TraditionalForm to see that .5 {-y, x}/(x^2 + y^2) can be use to express the vector function.

$$\left(\frac{-0.5y}{x^2+y^2},\frac{0.5x}{x^2+y^2}\right)$$

and actually,{-y, x}/(x^2 + y^2) is the gradient of the function ArcTan[y/x] or ArcTan[x,y]

Grad[ArcTan[x, y], {x, y}]
Grad[ArcTan[y/x], {x, y}] // Simplify
(* {-(y/(x^2 + y^2)), x/(x^2 + y^2)} *)


GraphicsRow[{VectorPlot[{-(y/(x^2 + y^2)), x/(
    x^2 + y^2)}, {x, y} \[Element] Disk[]], 
  VectorPlot[
   Grad[ArcTan[x, y], {x, y}] // Evaluate, {x, y} \[Element] Disk[]]}]

enter image description here

You can use .5 {-y, x}/(x^2 + y^2) // TraditionalForm to see that .5 {-y, x}/(x^2 + y^2) can be use to express the vector value function.

$$\left(\frac{-0.5y}{x^2+y^2},\frac{0.5x}{x^2+y^2}\right)$$

and actually,{-y, x}/(x^2 + y^2) is the gradient of the function ArcTan[y/x] or ArcTan[x,y]

Grad[ArcTan[x, y], {x, y}]
Grad[ArcTan[y/x], {x, y}] // Simplify
(* {-(y/(x^2 + y^2)), x/(x^2 + y^2)} *)


GraphicsRow[{VectorPlot[{-(y/(x^2 + y^2)), x/(
    x^2 + y^2)}, {x, y} ∈ Disk[]], 
  VectorPlot[
   Grad[ArcTan[x, y], {x, y}] // Evaluate, {x, y} ∈ Disk[]]}]

enter image description here

Source Link
cvgmt
cvgmt
  • 84k
  • 6
  • 97
  • 179

You can use .5 {-y, x}/(x^2 + y^2) // TraditionalForm to see that .5 {-y, x}/(x^2 + y^2) can be use to express the vector function.

$$\left(\frac{-0.5y}{x^2+y^2},\frac{0.5x}{x^2+y^2}\right)$$

and actually,{-y, x}/(x^2 + y^2) is the gradient of the function ArcTan[y/x] or ArcTan[x,y]

Grad[ArcTan[x, y], {x, y}]
Grad[ArcTan[y/x], {x, y}] // Simplify
(* {-(y/(x^2 + y^2)), x/(x^2 + y^2)} *)


GraphicsRow[{VectorPlot[{-(y/(x^2 + y^2)), x/(
    x^2 + y^2)}, {x, y} \[Element] Disk[]], 
  VectorPlot[
   Grad[ArcTan[x, y], {x, y}] // Evaluate, {x, y} \[Element] Disk[]]}]

enter image description here