Battle Mage Tradition

You learn additional spells when you reach certain levels in this class, as shown on the wizard table These spells count as wizard spells for you, but do not count against the number of wizard spells you know. 

Arcane Battle Study

You gained experiance through battles and being in combat as well as being in the library, studying spells.

Wizard level	spells
3rd	Healing Word , Magic Missile, Prism Sword, Chromatic Orb, Divine Favor, Magic Weapon,  Enhance Ability, Splash
5th	arms, armor, Lightning Bolt, [spells]Counterspell[/spell]
7th	Mordenkainen's Private Sanctum, Death Ward
9th	cone of cold, flame strike

Unconventional Casting, Battle Mage's Grace, and Acrobatics Daring Die

wizards who choose to pursue the school of the battle mage at level 3  must be focused on this fighting style exclusively such are disallowed from wearing any armor or shields regardless of availability.

 

Unconventional Casting
Your skills as an arcane duelist grant you proficiency with marital weapons and allow you to use any martial weapon as a spell-casting catalyst.

Battle Mage's Grace
When choosing the school of the battle Mage at 3rd level, your flare and charm in and out of combat allows you to add 3 plus your Charisma modifier as a unarmored bonus to your AC.

This feature disallows wearing any armor or shields regardless of availability.

Acrobatics Die
At 3rd level, you learn to perform impressive displays of martial prowess and speed. This power is represented by your Acrobatics die, the starting size of which is a d6. Once per round, you can expend Sorcery Points whenever you make an attack with a rapier to empower your strike in many ways using your Acrobatics die.

Changing the Die’s Size. When you reach certain levels in this class, the size of your Acrobatics die increases: at 6th level (d8), 10th level (d10),14th level (d12), and 18th level (d20).

Daring Acrobatics

• Corps-demure. At 3rd level, you can expend 1st level spell slot to cause extra damage from your Acrobatics die by lunging up to 20 feet towards a creature you can see as an Attack action. Roll to hit for the attack before lunging. If the attack hits, the target suffers extra force damage equal to the number you rolled on the Acrobatics die. If the attack doesn't hit, you still lunge towards the target but do not trigger opportunity attacks. 
  
  
• Moulinet. At 3rd level, as a bonus action, you can expend 1st level Spell slot to cause a weapon attack to deal extra force damage to a creature you hit and to any one additional creature that you can see within 5 feet of you. The additional creature only suffers the effects of the additional damage, not the initial weapon damage. The additional damage dealt to both creatures equals the number you roll on the Acrobatics die.
  
  
• Singularity. At 6th level, when taking the Attack option on your turn, you can expend 1 3rd level spell slot to gain the Extra Attack feature for this turn. This feature cannot be used again until you finish a short rest.
  
  
• Round-about. At 10th level, as a reaction, after attacking with a weapon, you can expend 2 4th level or higher spell slots to cause the weapon attack to deal extra force damage to the creature, propelling your dodge backwards. The extra damage equals half your class level (rounded up) times the number you roll on the Acrobatics die. Regardless if the attack hits or not, you jump backwards up to 5 feet times the number you roll on the Acrobatics die, but only on a hit does your movement not trigger opportunity attacks

Battle Mage's Spell Strike

at 6th level when a wizard chooses the school of the battle mage, they learn to infuse spells into there weapon. they can cast any spell they know that has one target, infusing it into their blade. This can be used in conjunction with spells that don't deal damage as well, such as command and hold person. If the Spell Strike critically hits, the spell damage is doubled, and saves made against any non damage spell have disadvantage.

When activating spell strike, you forego half the weapon damage for the casting of a spell. Roll a normal melee weapon attack against a single target and add your spellcasting ability modifier, while expending a spell slot of the level spell you are attempting to cast. On a hit, the target makes any relevant spell saves and/or you may roll damage for the spell and the weapon but you only do half as much damage for the weapon and the spell damage. One miss, the spell slot is expended, and the target it unaffected by the attack.  

This way of casting spells has its limitations, and as such, you may only use spell slots of level 5 or lower, and spells that target only a single creature.

Dual Cast and Balance Study

Dual-Cast

Starting at 10th level, your experimentation with spells allows you to weave different levels of spell-casting fluidly with each other. Whenever you cast a spell of 2nd or lower you can then cast a spell of any level 2nd and above that has a cast time of an Action as a Bonus action. Inversely, when using this feature to cast a spell of any level first, you can use your Bonus Action to then cast a spell of 2nd or lower instead. Roll to hit or against any associated saves/DCs accordingly for each spell. This feature can be used up to your one in addition to Intelligence modifier in addition to your proficiency bonus per long rest.

Balance

Starting at 10th level, you balance your magic and melee fighting style, allowing you to make one weapon attack as a Bonus Action whenever you cast a spell or cantrip as an Action and vice versa.

Regulated Spells and Acceleration with Perfect Balance

Regulated Spells

When you reach 14th level, you can create pockets of relative safety within the effects of your spells. When you cast any damaging spell using your Dual Cast or Balance feature, you can choose a number of creatures equal to 1 + the spell's level. The chosen creatures automatically succeed on their saving throws against the spell, and they take no damage if they would normally take half damage on a successful save.

Acceleration and Perfect Balance

Acceleration

Starting at 10th level, you have learned to weave swordsmanship and spell-casting together in perfect harmony. Whenever you successfully Dual-Cast a spell, you gain 1 point of Acceleration, which goes up to a maximum of (1 + Charisma Modifier). You can expand points of Acceleration to add additional effects to one of your abilities. These additional effects can be found in the Perfect Balance feature.

Perfect Balance

• Contre Sixte. After using Redoublement, expend at least 1 point of Acceleration (up to 3) to summon 2 spectral replicas of your weapon per point to attack the creature. Roll to hit and for damage once and use those rolls for all spectral weapons summoned.
  
  
• Embolden. As a bonus action you expend 1 point of Acceleration to increase the next Dual-Cast spells' damage by 1 additional damage die each. This feature cannot be used again until you finish a long rest.
  
  
• Maim and Mend. As a bonus action you expend all points of Acceleration to restore an equal amount of spell slots. This feature cannot be used again until you finish a long rest.