Help with Vector Function Calculations

In summary, the upside down triangle symbol indicates the del operator, which is used to take partial derivatives and can also be applied to vector functions as a dot or cross product to yield the gradient, divergence, or curl. It is analogous to the derivative operator d/dx.
  • #1
franky2727
132
0
missed a lecture and now have this homework problem and don't even know what the upside down triangle symbol indicates, can someone please give me a hand getting started, thanks

consider the vector function q=(1/4X^4 y^2 z, x^3 yz^6 - cosh(xz), 1/7x^3 z^7)

calculate f(x,y,z)=upside down triangle . q
 
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  • #2
That upside down triangle is the nabla symbol and is typically called "del". "Del" is an operator analagous to the derivative operator d/dx except that del takes partial derivatives. In Cartesian 3-space,

[tex]\boldsymbol{\nabla} \equiv
\hat{\boldsymbol x} \frac{\partial}{\partial x} +
\hat{\boldsymbol y} \frac{\partial}{\partial y} +
\hat{\boldsymbol z} \frac{\partial}{\partial z}
[/tex]

When applied to a scalar function f(x,y,z), the del operator yields the gradient of the function:

[tex]\boldsymbol{\nabla} f(x,y,z) \equiv
\hat{\boldsymbol x} \frac{\partial f(x,y,z)}{\partial x} +
\hat{\boldsymbol y} \frac{\partial f(x,y,z)}{\partial y} +
\hat{\boldsymbol z} \frac{\partial f(x,y,z)}{\partial z}
[/tex]

The operator definition of del looks like a vector. With a little abuse of notation, it can be applied to vector functions as a dot product (yielding a scalar) and a cross product (yielding a vector):

[tex]\boldsymbol{\nabla} \cdot \boldsymbol{f}(x,y,z) \equiv
\hat{\boldsymbol x} \frac{\partial f_x(x,y,z)}{\partial x} +
\hat{\boldsymbol y} \frac{\partial f_y(x,y,z)}{\partial y} +
\hat{\boldsymbol z} \frac{\partial f_z(x,y,z)}{\partial z}
[/tex]

and similarly for the cross product. The expression [itex]\boldsymbol{\nabla} \cdot \boldsymbol{f}(x,y,z)[/itex] is called the divergence of f(x,y,z) while [itex]\boldsymbol{\nabla} \times \boldsymbol{f}(x,y,z)[/itex] is called the curl.
 

1. What is a vector function?

A vector function is a mathematical function that takes one or more input variables and produces a vector as an output. It can be represented as a set of parametric equations, where each component of the vector is a function of the input variables.

2. How do I calculate the derivative of a vector function?

To calculate the derivative of a vector function, you can use the rules for vector calculus, such as the product rule and chain rule. You can also express the vector function in terms of its component functions and then calculate the derivatives of each component separately.

3. What is a unit vector and how is it related to vector functions?

A unit vector is a vector with a magnitude of 1 and is used to indicate direction. In vector functions, unit vectors are often used to represent the direction of a vector at a specific point along the function.

4. How can I determine the domain and range of a vector function?

The domain and range of a vector function depend on the input variables and the possible values they can take. To determine the domain, you can look at the restrictions on the input variables, such as inequalities or specific values. The range can be determined by examining the output vectors and their possible magnitudes and directions.

5. Can vector functions be graphed?

Yes, vector functions can be graphed in a three-dimensional coordinate system where the input variables correspond to the x and y axes, and the output vectors correspond to the z-axis. The resulting graph is a curve or surface in 3D space, depending on the number of input variables and components of the output vector.

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