Invariance Math Example 3

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Example 3

easy
Show that the expression x2+y2x^2 + y^2 is invariant under the transformation (x,y)↦(−x,−y)(x, y) \mapsto (-x, -y).

Solution

  1. 1
    Apply the transformation: replace xx with −x-x and yy with −y-y.
  2. 2
    (−x)2+(−y)2=x2+y2(-x)^2 + (-y)^2 = x^2 + y^2.
  3. 3
    The expression is unchanged — it is invariant under this transformation.

Answer

(−x)2+(−y)2=x2+y2 (invariant)(-x)^2+(-y)^2 = x^2+y^2 \text{ (invariant)}
A quantity is invariant under a transformation if applying the transformation leaves the quantity unchanged. Here, squaring eliminates the sign, making x2+y2x^2+y^2 symmetric under negation.

About Invariance

A property of a mathematical object that remains unchanged when the object undergoes a particular transformation or operation.

Learn more about Invariance →

More Invariance Examples

Example 1 easy

Show that the sum of the digits of a multiple of 9 is always a multiple of 9. Verify with [formula]

Example 2 medium

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Example 4 medium

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