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Unformatted text preview: defined implicitly be the function: (meaning not solved for z) Find the partial of z with respect to x. We will take the partial derivative with respect to x of both sides of the equation
Using the Chain Rule: Solve for Same logic holds for the partial with respect to y: x z y x
Make sure you know how to find
equation with respect to x. without the formula!! You need to differentiate both sides of the y Example:
Find Solve the above for to get: (Notice you would of got the same answer using the formula)
Another Example of Implicit Partial Differentiation:
Find where z is implicitly defined by: We are now in a situation just like 101A where we had which is just an application of the chain rule Solve for zx 12.6: Directional Derivatives and the Gradient
The Gradient of is denoted by and is defined as is read “nabla” or “del” and is a differential operator:
In General: Prove the Gradient Vector is Normal to the Tangent Plane for a surface defined implicitly by Proof:
Let Then (where k is a constant) is a Level Surface. Consider a curve on the level surface passing through the point Since the curve is on the surface it must satisfy the equation
Thus or Differentiate Both Sides of
The equation in compact form:
is a tangent vector to the curve Thus . So is lying in the tangent plane to the level surface. is Normal to the Tangent Plane We really have two vectors that are perpendicular to the tangent plane:
And (inward normal) (outward normal) The above fact is very useful to find the equation of the tangent plane when the surface is defined implicitly.
Find the equation of the tangent plane to the surface at the point . This surface is the level surface where Thus is the Tangent Plane at the point . Normal Line: Notice for a surface that is defined explicitly: We can rewrite it as Take the gradient:
We then get our old tangent equation formula from earlier on: (for the point ) We can use the same argument from above to show that the gradient is also perpendicular to a level curve (instead of a
surface like above)
Prove the Gradient Vector is Normal to the Tangent Line for a curve defined implicitly by Proof:
Let which represents a surface. Then Let (where k is a constant) is a Level Curve to that surface. be a parameterization of the level curve Thus the curve satisfies the equation
or Differentiate Both Sides of
In compact form:
is a tangent vector to the curve Thus . So i...
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This note was uploaded on 03/05/2014 for the course MATH 101c taught by Professor Loukianoff,v during the Spring '08 term at Ohlone.
- Spring '08