Help with Vector Function Calculations

[franky2727]

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

 

[D H]

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,

$$\boldsymbol{\nabla} \equiv<br /> \hat{\boldsymbol x} \frac{\partial}{\partial x} +<br /> \hat{\boldsymbol y} \frac{\partial}{\partial y} +<br /> \hat{\boldsymbol z} \frac{\partial}{\partial z}$$

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

$$\boldsymbol{\nabla} f(x,y,z) \equiv<br /> \hat{\boldsymbol x} \frac{\partial f(x,y,z)}{\partial x} +<br /> \hat{\boldsymbol y} \frac{\partial f(x,y,z)}{\partial y} +<br /> \hat{\boldsymbol z} \frac{\partial f(x,y,z)}{\partial z}$$

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):

$$\boldsymbol{\nabla} \cdot \boldsymbol{f}(x,y,z) \equiv<br /> \hat{\boldsymbol x} \frac{\partial f_x(x,y,z)}{\partial x} +<br /> \hat{\boldsymbol y} \frac{\partial f_y(x,y,z)}{\partial y} +<br /> \hat{\boldsymbol z} \frac{\partial f_z(x,y,z)}{\partial z}$$

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