Real Valued Functions on R^3 - Chain Rule ....?

In summary, the conversation discusses using the chain rule for multiple variables in Exercise 4(a) of Section 1.1 in Barrett O'Neil's book on Elementary Differential Geometry. The suggested function is $g: \Bbb R^3 \to \Bbb R^3$ and the chain rule formula is $Df(\mathbf{p}) = Dh(g(\mathbf{p}))Dg(\mathbf{p})$, with the LHS being a 1X3 matrix and the RHS being the product of a 1X3 matrix and a 3X3 matrix.
  • #1
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I am reading Barrett O'Neil's book: Elementary Differential Geometry ...

I need help to get started on Exercise 4(a) of Section 1.1 Euclidean Space ...

Exercise 4 of Section 1.1 reads as follows:View attachment 5186Can anyone help me to get started on Exercise 4(a) ...

I would guess that we need the chain rule for multiple variables but how do we formulate the dependencies of the functions involved and what is the correct form of the chain rule to use ...

Help will be appreciated ...

Peter
 
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  • #2
I would use the following function:

$g: \Bbb R^3 \to \Bbb R^3$

given by $g(x,y,z) = (g_1(x,y,z),g_2(x,y,z),g_3(x,y,z))$

Then $f= h \circ g$ and the chain rule gives:

$Df(\mathbf{p}) = Dh(g(\mathbf{p}))Dg(\mathbf{p})$

The LHS will be a 1X3 matrix, and the RHS will be the matrix product of a 1X3 matrix and a 3X3 matrix.

Each ENTRY of the LHS (which will be your partials $\dfrac{\partial f}{\partial x_i}$) will be a dot product.
 

FAQ: Real Valued Functions on R^3 - Chain Rule ....?

What is the chain rule for real-valued functions on R^3?

The chain rule for real-valued functions on R^3 is a mathematical principle used to calculate the derivative of a composite function. It states that the derivative of a composite function is equal to the derivative of the outer function multiplied by the derivative of the inner function.

How is the chain rule applied in real-life situations?

The chain rule is commonly used in physics, engineering, and economics to analyze and solve complex systems. For example, in physics, the chain rule is used to calculate the acceleration of an object moving in a curved path by breaking it down into smaller components.

Can the chain rule be extended to higher dimensions?

Yes, the chain rule can be extended to higher dimensions. In fact, the chain rule can be applied to any function with multiple inputs and outputs, regardless of the dimension.

What is the significance of the chain rule in multivariable calculus?

In multivariable calculus, the chain rule is an essential tool for calculating partial derivatives of multivariable functions. It allows us to break down a complex function into simpler parts, making it easier to analyze and differentiate.

Are there any special cases where the chain rule does not apply?

There are certain cases where the chain rule does not apply, such as when the function is not differentiable or when the inputs and outputs are not real numbers. Additionally, the chain rule may not apply if the function is discontinuous or has sharp corners.

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