# 矢量微积分在流体力学中的应用代写|VECTOR CALCULUS WITH APPLICATIONS IN FLUID MECHANICS MATH225 University of Liverpool Assignment

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## Instructions:

Vector calculus is an important branch of mathematics that deals with the study of vector fields and their associated operations. The main vector operations studied in vector calculus are the gradient (grad), divergence (div), and curl. These operations are used to study various physical phenomena in fields like fluid mechanics and electromagnetism.

The gradient is a vector operator that describes the rate of change of a scalar field in a particular direction. Mathematically, the gradient of a scalar field is defined as the vector obtained by taking the partial derivative of the scalar field with respect to each of its variables. The gradient of a scalar field is a vector field, and it points in the direction of the maximum rate of increase of the scalar field.

The divergence is a scalar operator that measures the rate at which a vector field flows out of a given point in space. Mathematically, the divergence of a vector field is defined as the sum of the partial derivatives of its components with respect to each of the coordinates. The divergence of a vector field is zero if the field is “solenoidal” or “incompressible,” meaning that it does not have sources or sinks.

The curl is a vector operator that describes the rotational behavior of a vector field. Mathematically, the curl of a vector field is defined as the vector obtained by taking the cross product of the gradient of its components with respect to each of the coordinates. The curl of a vector field is zero if the field is “irrotational,” meaning that it does not have any swirl or vortices.

The motion of a point $P$ is given by the position vector $\vec{R}=3 \cos t \hat{\mathrm{i}}+3 \sin t \hat{\mathrm{j}}+t \hat{\mathrm{k}}$. Compute the velocity and the speed of $P$.

a) Find the area of the space triangle with vertices $P_0:(2,1,0), P_1:(1,0,1), P_2:(2,-1,1)$.

a) $\overrightarrow{P_0 P_1} \times \overrightarrow{P_0 P_2}=\left|\begin{array}{rrr}\hat{\imath} & \hat{\jmath} & \hat{k} \ -1 & -1 & 1 \ 0 & -2 & 1\end{array}\right|=\hat{\imath}+\hat{\jmath}+2 \hat{k} . \quad$ Area $=\frac{1}{2}\left|\overrightarrow{P_0 P_1} \times \overrightarrow{P_0 P_2}\right|=\frac{1}{2} \sqrt{6}$.

c) Find the intersection of this plane with the line parallel to the vector $\vec{V}=\langle 1,1,1\rangle$ and passing through the point $S:(-1,0,0)$.

c) Parametric equations for the line: $x=-1+t, y=t, z=t$.
Substituting: $-1+4 t=3, t=1$, intersection point $(0,1,1)$.